Talk:Topology/Archive 2

Archive 1

Archive 2

Useability

Can anyone simplify parts of this article? I can't, reading the intro & history, say that I gained any understanding. ..And I come with B.S. Physics from Carnegie-Mellon. Pitty the more novice! --Ej0c 10:54, 5 June 2006 (UTC)

I think that the sentence "Two figures that can be deformed one into the other are called homeomorphic" in the introduction is somewhat inaccurate. Really this is some kind of homotopy concept, not homeomorphism. On the other hand, the "deformation" concept is far more useful for someone who has little mathematical background. I'm not sure what the best solution to this is. --Dmharvey 03:55, 27 May 2005 (UTC)

new intro

The articles on Topology, Topological Glossary, and Topological Space currently do not clearly define the areas they cover. I've rewritten the intro here to try to help the beginner find her way around these various articles. Rick Norwood 19:07, 24 October 2005 (UTC)

The idea that topology has usurped the old place of geometry in mathematics, cuckoo-style, was 'big in the sixties'. It reads oddly now. Those guys - well, let's just say POV and have done. The idea that topology is basically the Erlangen program for the homeomorphism group: that probably dates back to the 1930s, but in a sense the functor concept revealed that it was just part of the picture (the category of topological spaces as a whole is the real object of study). There was the division algebraic topology, differential topology, geometric topology that I think dates from differential topology being defined in the 1950s. A palace revolution made geometric topology the most fashionable rather than the least (Thurston, knot polynomials ... late 1970s/early 1980s). It would be good to get an intro that didn't just take the clichés at face value. Charles Matthews 21:21, 24 October 2005 (UTC)

As it reads now, the intro is a bit vague for non-mathematicians. Would it make sense to add something like:

Loosely speaking, topology concerns itself with which points in a space are 'next to' one another without concern for distance per se. A topology is a defined system of adjacencies for a space; for example, a road network could have a walking topology different from a driving topology due to one-way streets.

Is that correct? If so, I think it would help to add something like this to shed light on why it is useful to think about spaces under arbitrary continuous deformation (i.e., spaces in which topology matters but metric doesn't). —Ben FrantzDale (talk) 19:41, 11 April 2011 (UTC)

I don't think that would help at all. It would make people think it had something to do with graphs. What's there is a reasonable description for non-mathematicians I think. Personally I would say rather than that it is concerned about spatial properties that are preserved under continuous deformations, instead that it is based on a study of continuous functions and originated in the study of the properties of spatial deformations that involve stretching but no tearing or gluing. — Preceding unsigned comment added by Dmcq (talk • contribs) 21:22, 11 April 2011

BenFrantzDale is talking about a different sense of the word topology, one used in computer science. It's somewhere in topology (disambiguation), which is hatnoted at the top, but I'm not sure that's really enough.

On a similar note, I've long thought that we should have an article called topology (object) or some such, for the set of open sets. Topology (disambiguation) has a link to topological space, but that isn't quite the same thing, because it includes the underlying points as well as the opens. Often you want to talk about different topologies on the same space. This meaning, I think, is also important enough to be called out in a separate hatnote, and not folded into the disambig page. --Trovatore (talk) 23:25, 11 April 2011 (UTC)

New To Advanced Math

Hi; I'm trying desperately to understand many of these advanced principals of mathematics, such as lattice groups, but no matter how many times I review the material, it doesn't sink in. Could someone please provide examples, problems to solve (with their solutions) and/or ways to visualize this? beno 26 Jan 2006

I've offered you a few suggestions over in the Topological Space article. Rick Norwood 22:48, 26 January 2006 (UTC)

Mandelbrot?

Why does the history have so much from Mandelbrot, who is not a topologist? Charles Matthews 23:14, 31 January 2006 (UTC)

I was sort of wondering that, too. --Trovatore

I don't think that quote adds anything essential. I'm removing it. --C S (Talk) 22:26, 15 February 2006 (UTC)

Demarcation of branches of topology

So I noticed that there were a few places (topology, general topology, Category:Topology, Category:General topology; did I miss any?) where there was text that implied that general topology was an elementary discipline. I think maybe 20 years ago there was a common opinion that general topology had been more or less mined out, but I think it's fair to say that there's been a considerable resurgence since then, and the new stuff is not elementary at all.

However I'm not entirely sure I picked the right dividing line. What I said was that the other topological disciplines needed something like a manifold structure. Is that true? I know only about their applications to manifolds or near-manifolds, but that may just be a reflection of the fact that I don't really know much about those disciplines. --Trovatore 00:07, 1 February 2006 (UTC)

Yes, there's a distinction to clarify. There is the pedagogic subject one has to learn, in which topological spaces are introduced and continuity defined in general. This area, which is foundational for things like functional analysis and manifolds, was pretty much there by the 1940s, with things like filters and uniform spaces as refinements. Then there is the research area also called 'general topology' which grew out of the concerns of the 1930s (dimension theory, Polish school and descriptive set theory, Texas topology), which continues. But there is not a huge cross-over, is there? An old text like Kelley is still OK for graduate students. Charles Matthews 09:45, 1 February 2006 (UTC)

So I don't think I'd include descriptive set theory as part of general topology; descriptive set theory is more about definability than it is about continuous functions. Not sure what you mean by "cross-over". I think the pedagogic subject more or less blends into the research subject and don't see a need to make a demarcation at the interface between those two. My question still stands: Do the other branches need a manifold structure or something close to it? If so, then I think my text is fine.

As an example, a general topologist might find huge distinctions between two totally disconnected spaces and write entire books on different kinds of totally disconnected spaces, but am I right that he's the only sort of topologist who'd be interested in them? To an algebraic topologist, such spaces live in the category of "pathologies that I don't want to know about and have no tools to address if I did", is that fair? --Trovatore 14:46, 1 February 2006 (UTC)

Well, no, totally disconnected spaces are of considerable interest (for example the p-adic integers). As Stone spaces they have a high tendency to be homeomorphic, though. They come to be interesting approximately because they can carry p-adic analytic manifold structure. There are algebraic topologists who think about profinite things. On the distinction, there is an ambiguity in general topology, since general might connote 'of general use', as much as 'of great generality'. Back in the history, there really was a time before the logicians got hold of descriptive set theory. Charles Matthews 15:13, 1 February 2006 (UTC)

Wow, that sounds really interesting—I never knew there was a category "p-adic analytic"; guess I never really thought about it. Still, that fits with my rough and ready distinction of "carrying something like a manifold structure". Of course I'm trying to get at the "of great generality" meaning; I think that's the appropriate one to use to characterize general topology. Do you think my attempt in the four places I listed looks reasonable? --Trovatore 15:52, 1 February 2006 (UTC)

• JA: Kelley is a wonderful classic, not just for Topology alone but as a guide to a Golden Age style of writing, and several generations of mathematicians learned the only formal set theory they ever had out of its appendix. But the standard definitions of several concepts, like the Product Topology, changed after it was written, so it no longer works except as elective reading for graduate courses. Jon Awbrey 13:04, 1 February 2006 (UTC)

Hmmm ... is the thing about the product topology really true? Charles Matthews 13:07, 1 February 2006 (UTC)

No. In addition, the book is still cited as a standard reference and used in graduate courses (although certainly not as popular as it once was). I doubt things have changed very much after the book's publication as the book is not that old and was very modern when it first came out. My readings of the book suggest to me that much of it is still very standard stuff. --C S (Talk) 00:19, 15 February 2006 (UTC)

• JA: I will check when the sun comes up in the library. Please be gentle with my memory, I know I have to, I took Topology courses intermittently over a span of 15 years and it was a time of transition, what time isn't, so I may have mixed up the Box and the Product yet again, or some other misremember. Jon Awbrey 13:14, 1 February 2006 (UTC)

On being elementary

• JA: I've been noticing this widespread misunderstanding here in WikioPolis as to the general meaning of the word "elementary" in mathematics. It all goes back to Euclid's Stoicheia (singular Stoicheion), a Greek word for many things, from the shadow of a gnomon on a sundial, to letters and numbers, to steps of a stair or stepping stones, to the stars, but in geometry having the specific senses glossed below:

Stoicheion. 3. the elements of proof, e.g. in general reasoning the prôtoi sullogismoi, Arist.Metaph.1014b1; in Geometry, the propositions whose proof is involved in the proof of other propositions, ib.998a26, 1014a36; title of geometrical works by Hippocrates of Chios, Leon, Theudios, and Euclid, Procl. in Euc.pp.66,67,68F.: hence applied to whatever is one, small, and capable of many uses, Arist.Metaph.1014b3; to whatever is most universal, e.g. the unit and the point, ib.6; the line and the circle, Id.Top.158b35; the topos (argument applicable to a variety of subjects), ib.120b13, al., Rh.1358a35, al.; stoicheia ta genê legousi tines Id.Metaph.1014b10; (Liddell & Scott, A Greek-English Lexicon)

• JA: See also: Euclid, Elements, Thomas L. Heath (ed.) — Online at Perseus

• JA; Jon Awbrey 04:52, 1 February 2006 (UTC)

On being famous, on being cited, on being not

• JA: Can't say either way about Munkres, but J.L. Kelley definitely deserves an article. NB. It is necessary to distinguish "works cited" from "standard lit" somehow or other. Most journals, monographs, and surveys that I know mark the distinction as "references" vs. "bibliography", respectively. I've have noticed folks hereabouts have some beef about Bibs, so I've been trying several other terms, like "further reading" or "literature". Jon Awbrey 04:15, 2 February 2006 (UTC)

On Oleg's comment in the edit summary about de-linking Munkres and Kelley as they are not famous enough or something like that...I have to disagree. They more than meet the standard of notability I often see on mathematicians. Munkres' notoriety as an author is probably already enough, but he has done important work, e.g. on smoothing PL-structures (for which I've often seen him cited). Kelley would certainly meet notability criteria just on the basis of his influence in the mathematical community (see obituaries of him for more details). Also, Kelley's book is also very famous, and he proved the theorem that AC is equivalent to Tychonoff's theorem. Articles on these people are certainly welcome and hopefully someone will do one in the future. --C S (Talk) 00:28, 15 February 2006 (UTC)

Yes I agree that both authors (especially Kelly), are notable enough for there own article, in fact creating an article on Kelly, has been on my todo list for some time. ;-) Paul August ☎ 18:15, 5 June 2006 (UTC)

And now he's been done, but not by me! Paul August ☎ 20:08, 17 July 2006 (UTC)

Not totally sure about Munkres (others are probably better placed to comment) but I would strongly support keeping Kelly. I believe the influence his book has had on mathematicians of a certain generation (my generation, probably ...) is in itself enough to entitle him to be considered notable. Madmath789 18:41, 5 June 2006 (UTC)

I was in undergrad 88-92 and grad 93-97 and Munkres was the standard book. Kelly was the standard harder reference. They both are clear cut includes. jbolden1517^Talk 18:46, 5 June 2006 (UTC)

Stub type for topology

Just a heads-up that there's a new stub type, {{topology-stub}}. Assistance in correctly classifying existing articles would be great, and I also wanted everyone to know it's there for new articles. --Trovatore 19:59, 7 February 2006 (UTC)

String Theory

Doesn't the Donut and coffee cup example represent something in string theory? 70.111.251.203 15:19, 7 March 2006 (UTC)

It is a standard example in topology. And string theory is part of topology, so a book on string theory might mention it. But it is an elementary example, and string theory is anything but elementary. Rick Norwood 21:52, 7 March 2006 (UTC)

String theory is not part of topology... Dylan Thurston 05:07, 17 August 2007 (UTC)

this sentence is tricky

In this sense, a topology is a family of open sets which contains the empty set and the entire space, and is closed under the operations union and finite intersection.

Maybe finite intersection means Finite intersection property, but operations union definitely is not Operation Union. Can someone try to explain operations union? --Abdull 19:10, 8 June 2006 (UTC)

The word "operations" in this sentence is designed to introduce the names of the two operations that a topology must be closed under. They are the operations of union and finite intersection. A similar wording is used in the statement that the set of integers is closed under the operations addition and multiplication. It is clear, now, or should that sentence be rewritten? Rick Norwood 19:18, 8 June 2006 (UTC)

Lacan

Should Lacan's "topology" really be in this article? It doesn't seem any more related to topology than the new age "energy" has to do with energy in physics. The physics energy article doesn't have a "Energy in Reiki" section does it? That would belong in the Reiki article not the energy (physics) article, likewise I'm inclined to think that Lacan's "topology" belongs in the article about Lacanian psychoanalysis, not this article. Brentt 12:09, 16 August 2006 (UTC)

{c,f,h,k,l,m,n,r,s,t,u,v,w,x,y,z} are only homotopic not homeomorphic

The article mentions that "topological equivalence" formally means homeomorphity, and then as an example gives that {c,f,h,k,l,m,n,r,s,t,u,v,w,x,y,z} are in one class of equivalence. May be too pedantic a notice, but {c,f,h,k,l,m,n,r,s,t,u,v,w,x,y,z} are not homeomorphic, but only homotopic. Shouldn't this better be fixed somehow? Tamokk 20:47, 20 August 2006 (UTC)

It has been fixed with the assumption that the lines of the letters are assumed to have non-zero width. --TheVelho 02 January 2007

You can't fix a bad idea by doing the bad idea well. The idea of using letters to represent surfaces with boundary -- as this section does -- is a colossally bad idea if your goal is to communicate to the non-topologist what topology is all about. The inclusion of the phrase "the lines of the letters are assumed to have non-zero width" does keep this colossally bad idea from being technically false, but it is a huge didactical mistake. It would be far, far more useful for communicating the idea of topological equivalence to classify uppercase, sans-serif letters according to topological type.Daqu 13:41, 27 February 2007 (UTC)

I have to wholeheartedly agree with this! I missed the nonzero width caveat on the first reading and naturally assumed the classification was up to homotopy equivalence. I think it's quite tricky for the layman to see how certain thick objects are the same. Anyway, I'm going to do some edits in the near future to change this to a classification of uppercase sans-serif letters as you suggest, up to both homotopy equivalence and homeomorphism. I think it's better to include figures so we don't get caught up in the font business (which I'll get to...). --C S (talk) 10:25, 21 July 2008 (UTC)

I don't think H and K are homeomorphic, K can be divided into 4 components by removal of a point H cannot be. 128.180.52.123 (talk) 01:53, 1 September 2009 (UTC)

I think that the idea is that the point where the two diagonals meet is not the same point where the upper diagonal meets the vertical. But this is still problematic. If the letters are understood as having nonzero thickness (and nothing in the text says they're not, unless I missed it), then H and K are both just disks, as are 17 other letters. I'm less than convinced that any version of this is really a good idea; if it's to be kept it needs to be explained more carefully, and I'm not sure how useful it'll be once that explanation is added. --Trovatore (talk) 02:18, 1 September 2009 (UTC)

It's 2011 and this is still horribly misleading. Letters are not zero-width, nor is the assumption clearly specified. Since homotopy and homeomorphism are indistinguishable on letter with non-zero width, I'd suggest a different example altogether. —Preceding unsigned comment added by 129.83.31.1 (talk) 13:46, 1 March 2011 (UTC)

Topology in psychoanalysis

I'm curious as to why this section is even here. So a psychologist has borrowed some words from mathematics. Mathematicians do that all the time. This seems to be the case of Undue weight, is your average reader of this article interested? No. The correct weighting for this is 0 - delete. --Salix alba (talk) 14:33, 12 September 2006 (UTC)

As my last edit summary shows, not fifteen minutes ago I was of the same opinion, but a bit too cautious to just get rid of it myself. It is, after all, bollocks. (And I didn't notice that after Anti-Vandal Patrol's revert of the IP's edits, the IP himself removed the paragraph, and my supposed "revert to IP" actually had the effect of restoring it).

However, on the other hand, a good paragraph which tells the reader precisely this (that Lacan's topology is bollocks) might be in order. Don't worry, I'm not going to start any revert wars over this, but I do think that there is something to be said for a five line paragraph mentioning Lacan's ramblings and pointing out to the reader that they are not to be confused with the actual subject of topology, but are merely the pretentious maunderings of yet another postmodernist. Just to alleviate any confusion that might exist - and after all, a reader might be drawn to this article to find precisely that fact out... Byrgenwulf 14:50, 12 September 2006 (UTC)

It might be better under the Jacques Lacan page, or Borromean rings, or maybe a new page Mathematical models in psychoanalysis, or maybe we could broden the section to topological methaphors, I'm sure there are many other instances where the languare of topology has been borrowed in other fields. Off the top of my head we have the Tokamak torus in physics, which works because of the Hairy ball theorem. Thom's work comes to mind, while not strictly topology he was very keen on aplying mathematical ideas to diverse phenomena. I saw a book yesterday which had a chapter on postmodern mathematics which this stuff probably fits.

It's actually not complete bolocks! What lucan is saying is that the Real, the Imaginary, and the Symbolic are interlinked and if you remove one the other two fall apart. He's used pictorial means to represent this, theres quite a few similar instances in the psychological field where they use some form of pictorial representation to illustrate relationships of abstract concepts. By the methodology of psychology its about as valid as much of the other work and by the subjects nature they need to use a very different methodological framework. We in mathematics are not equipped to judge, apart from say this is not mathematics. --Salix alba (talk) 15:37, 12 September 2006 (UTC)

Saying that the real, imaginary and symbolic are interlinked is fine, as would be using a diagram to illustrate the idea. However, saying that "the phallus is the square root of minus one" (as Lacan did) because of the result of a so-called "calculation" is out and out pretentiousness...in other words, using terminology to sound posh but actually not saying very much of any import. I certainly appreciate the benefits of using analogies to and from maths, it's just that Lacan didn't use his "method" as analogy: he seemed to think he was actually doing mathematics. So do his disciples. But he wasn't: he was using mathematical words. There is one Hell of a difference.

But fine, the paragraph can stay out. I suppose the article is better without mention of him... Byrgenwulf 15:59, 12 September 2006 (UTC)

"What should we make of Lacan's mathematics? Commentators disagree about Lacan's intentions: to what extent was he aiming to 'mathematicize' psychoanalysis? We are unable to give any definitive answer to this question — which, in any case, does not matter much, since Lacan's 'mathematics' are so bizarre that they cannot play a role in any serious psychological analysis.

"To be sure, Lacan does have a vague idea of the mathematics he invokes (but not much more). It is not from him that a student will learn what a natural number or a compact set is, but his statements, when they are understandable, are not always false. On the other hand, he excels (if we may use this word) at the second type of abuse listed in our introduction: his analogies between psychoanalysis and mathematics are the most arbitrary imaginable, and he gives absolutely no empirical or conceptual justification for them (neither here nor elsewhere in his work).

"[...] Lacan's defenders (as well as those of the other authors discussed here) tend to respond to these criticisms by resorting to a strategy that we shall call 'neither/nor': these writings should be evaluated neither as science, nor as philosophy, nor as poetry, nor ... One is then faced with what could be called a 'secular mysticism': mysticism because the discourse aims at producing mental effects that are not purely aesthetic, but without addressing itself to reason; secular because the cultural references (Kant, Hegel, Marx, Freud, mathematics, contemporary literature ...) have nothing to do with traditional religions and are attractive to the modern reader."

Quoted from Alan Sokal and Jean Bricmont, Intellectual Impostures (2nd edition, 2003), p. 34.

There are so many more important things which should be discussed in a general overview of topology than Lacan's misappropriation of its terminology. Anville 18:22, 12 September 2006 (UTC)

Good point Anville. Its just not appropriate subject matter for this article period. Its not about whether its bull or incredibly insightful, its simply that Lacan has not contributed anything of note to the study of topology, which this article is about. Brentt 19:46, 12 September 2006 (UTC)

toroid example is odd

It seems to me that the "toroid" shown has many holes! Either the picture should be improved, or the reader should be instructed that the picture is meant to be interpreted as something like a wire mesh under the surface of the real toroid. --69.70.139.251 01:19, 20 October 2006 (UTC)

horribly discouraging intro

that intro is full of jargon that nobody is going to understand

Your criticism is entirely apt. I'm going to try to work on the intro, and see what happens. By the way, sign your posts with four tildes. Rick Norwood 14:19, 24 October 2006 (UTC)

question

quote: "......For example, the square and the circle have many properties in common: they are both one dimensional objects and both separate the plane into two parts, the part inside and the part outside......"

I always thought square and circle are 2 dimensional (length and width) if someone could enlighten me please. thank you

The circle and the square are the outline alone, not the area they contain. They're infinitelly thin objects, and that's why it was said they divide the plane in two parts "the inside and outside". At least this is what I understood from that paragraph. ☢ Ҡi∊ff⌇↯ 06:49, 11 November 2006 (UTC)

Laypersons use the word "circle" in two ways, so that a hoop is a circle but so is a plate. Mathematicians need to distinguish between these two objects, and so the hoop is a circle, the plate a disk. In technical terms, a circle is the set of all ordered pairs (x,y) such that the sum of the squares of the variables is constant.Rick Norwood 13:17, 11 November 2006 (UTC)

Of squares and circles

Those who have this page on their watchlist will have seen various edits relating to what dimension the sqaure and the circle are. I will try and put this matter to bed:

1. Points (and finite collections thereof) are zero-dimensional.
2. Lines of any length (and finite unions thereof) are one-dimensional.
3. In plane geometry, "solid" (i.e. filled-in) regions of the plane of any size are two dimensional.
4. In geometry of three or more dimensions, planes of any size (and finite unions thereof) are two dimensional
5. The square and the circle are not points, and are therefore, not zero dimensional.
6. The square and the circle refered to in the article "both separate the plane into two parts, the part inside and the part outside"
7. The shapes been reffered to are *not* the filled in versions; therefore they are not two dimensional.
8. The shapes are in fact both a short line bent into the relevant shape, which then has its ends joined togther.
9. Therefore a square and a circle are of the same dimensionality as a line
10. Therefore a square and a circle are one dimensional

There are various other arguments I could give (for instance, I can descrie a square or a circle using a function of one variable). I'd also like to point that a filled in circle is a disc (which is a 2D object).

If you don't understand or disput any part of my reasoning, please say indicate which point number is the problem, and I will do my best to explain. Tompw (talk) 23:41, 25 December 2006 (UTC)

This is basically what I argued on a comment in this very page a few months ago. I wonder if we could changed it from:

For example, the square and the circle have many properties in common: they are both one dimensional objects and both separate the plane into two parts, the part inside and the part outside.

To:

For example, the square and the circle have many properties in common: they are both, topologically, one dimensional objects and both separate the plane into two parts, the part inside and the part outside.

The confusion arises because people have in mind the geometric view of a circle. As you probably know, in geometry the number of coordinates is used to give it a dimension, but in topology, we are talking about the dimension of the object itself. So, for example, we have a geometric 2-sphere being the same as the topologic 1-sphere.

I took the liberty to add this bit into the article. Feel free to make it clearer (we could add a little note at the bottom of the article, but I don't quite know the notation for those) — Kieff 00:30, 26 December 2006 (UTC)

An excellent addition :-) (To be precise, the circle and square are one-dimensional objects that are generally seen as a 2D immersion). (More geomtetric, but less rigrourous way to see that a cirlce is 1D: you can describe it with an equation of one variable). Tompw (talk) 22:28, 26 December 2006 (UTC)

---

Just a side note here: Claim (1) at the top of the section is not quite right. There are lots of zero-dimensional spaces other than finite collections of points. For example Cantor space is zero-dimensional --Trovatore 06:13, 26 December 2006 (UTC)

Yes, you're absolutly right. However, I felt that would be irrelevant to the argument I was making. (I've removed the word "only" in point 1... also the Cantor set is a countably infinte collection of points. Of the top of my head, I think any countably infinite collection of points is 0D, but I definately didn't want to get into coutnable vs. uncountable infinites, because a line is an (uncountable) infinte collection of points). Tompw (talk) 22:28, 26 December 2006 (UTC)

Cantor space is actually uncountable -- in fact it has the same cardinality as the reals.

I think a countable subset of R (or more generally Rⁿ is always zero-dimensional. The result certainly does not hold for topological spaces in general, though. Consider a topological space with just three points; call them a, b, and c. Let the open sets be:

• The empty set
• {a}
• {a,b}
• {a,c}
• {a,b,c}

You can easily check that this is a topology, and that the space is not zero-dimensional (for example the open cover given by {a,b} and {a,c} has no refinement into open sets that do not meet -- actually, it has no proper refinement at all). --Trovatore 08:20, 28 December 2006 (UTC)

That's neat :-) What is the dimension of that one? Actually, doesn't it depend on what sort of dimension you use? (Perhaps I should have said I was talking about subsets of Rⁿ) Tompw (talk) 18:42, 2 January 2007 (UTC)

The cantor set has (Hausdorff) Dimension log(2)/log(3). Seems strange to call it zerodimensional. I don't know other dimensions which make any sens for the cantorset.

Informal comment in the article

Is the clause "(Of course, the topologist must be an American. British doughnuts don't have a hole)" really necessary. It seems very informal and irrelevant to me. Davw 11:30, 16 January 2007 (UTC)

You're right, it's irrelevant. I've decided to remove it from the article by commenting it out, just in case someone comes in the future trying to add it in. This sort of nitpicking is pretty common over here. — Kieff 12:12, 16 January 2007 (UTC)

Erroneous characterization of toroid graphic

The caption beneath the (Mathematica?) graphic of a "toroid" reads as follows: A toroid in three dimensions; A coffee cup with a handle and a donut are both topologically indistinguishable from this toroid.

This is blatantly untrue. The "toroid" as shown is a depiction of a torus -- a 2-manifold, i.e., a surface. But the coffee cup and the doughnut are 3-manifolds with boundary. (Yes, indeed, the coffee cup and doughnut are classical examples of two different shapes that are topologically equivalent: each one is topologically a solid torus. But neither one is a torus per se.) This is not a minor point; this error will be rather confusing to a newcomer. It needs to be corrected.Daqu 13:12, 27 February 2007 (UTC)Daqu 13:20, 27 February 2007 (UTC)

Gender

Where the gender is indeterminate the masculine is used. To specifically use the feminine means it can only be applied to females. Maybe English isn't your first language, or you're just not very good at it. —The preceding unsigned comment was added by 83.70.229.193 (talk) 23:47, 10 April 2007 (UTC).

Maybe you've been in a cave since the 1970s? It has become commonly accepted practice to use either "he" or "she" for neutral gender. Either your simply trying to make a political statment or you are honestly ignorant of the change in commonly accepted practice. (Maybe you havn't read much of anything that was published after 1980?) As per wikipedia assume good faith policy I'll assume your just ignorant of the change and not using wikipedia as a soapbox for you political beliefs. Check any number of modern publications: you'll find the practice of alternating almost as often as not. Brentt 02:18, 11 April 2007 (UTC)

I don't know if I take quite as strong a p osition as Brentt on this, but it should be noted that the American Heritage Dictionary comments in the entry "she":

USAGE NOTE: Using she as a generic or gender-neutral singular pronoun is more common than might be expected, given the continuing debate regarding the parallel use of he. In a 1989 article from the Los Angeles Times, for instance, writer Dan Sullivan notes, “What's wrong with reinventing the wheel? Every artist has to do so in her search for the medium that will best express her angle of vision.” Alice Walker writes in 1991, “A person's work is her only signature.” It may be argued that this usage needlessly calls attention to the issue of gender, but the same argument can be leveled against generic he. This use of she still carries an air of unconventionality, which may be why only three percent of the Usage Panel recommends it in sentences like A taxpayer who fails to disclose the source of &rule3m; income can be prosecuted under the new law. •Some writers switch between she and he in alternating sentences, paragraphs, or chapters. This practice has been gaining acceptance, especially in books related to fields like education and child development, where the need for a generic pronoun is pervasive. It can also be seen in academic journals, where the sentence The researcher should note that at this point in the experiment she may need to recheck all data for errors might be followed later in the same section by The researcher should record his notes carefully at this stage. This style may seem cumbersome, but if generic pronouns are required, alternating between she and he can offer a balanced solution in an appropriate context. See Usage Notes at he1, they.

Of further relevance is the AHD entry on "he", which is quite extensive, but concludes "The writer who chooses to use generic he and its inflected forms in the face of the strong trend away from that usage may be viewed as deliberately calling attention to traditional gender roles or may simply appear to be insensitive."

Where does this leave us, as Wikipedia editors? Rather than name-calling (as I see no evidence the regular editors of this page are in fact inept users of English, quite the contrary!), let me suggest a common practice -- stick with the usage style given by creators of the article, i.e. those that have contributed the most to this article.

The origin of this well-known joke in this article goes back to the very early days of this article, March 11 2003 see diff, where the pronoun "she" was used as a generic pronoun. Since then, it has been reworked with much of the article, to remain there relatively undisturbed until this last year, when one anon (who, like the current anon, apparently has made no other contributions to this article), insisted on more than once changing the "she" to "he", prompting Oleg Alexandrov to put a html comment about it in the article. --C S (Talk) 03:24, 11 April 2007 (UTC)

Indeed, I noted at the anon's talk page that I'd seen this joke in the history back to April 2004. But it is also unfortunate that the use of 'she' coincides with a joke. The stereotypical rarity of women mathematicians combined with the humorous setting can make some uncomfortable. But I have more of a problem with those who are humor-impaired, such as the editor who was offended by the Haldane quote "God must have an inordinate fondness for beetles" being in the article on beatles. Anyway, I have copped-out, avoided the whole issue, by using the singular 'they', so that everyone can agree "that's wrong!" Shenme 04:03, 11 April 2007 (UTC)

Well males and females are topologically equivalent, so I can't understand the fuss. (sorry couldn't resit) --Salix alba (talk) 16:29, 11 April 2007 (UTC)

Just to throw in my two cents, I have no trouble using female pronouns and I actually advocate this view as a way of compensating for the male-dominant practice of using the male pronoun everywhere. I say, change them all to female! Of course, we all know that alternating is probably the best thing to do...but in my personal publications I will continue to use all female until we really reach a point of balance. Cazort (talk) 15:36, 2 March 2008 (UTC)

WTF

So let me get this straight, you want to use an excessively over the top politically correct bastardisation of the English language to make a sexist derogatory joke about women mathematicians. I'd suggest a change in your medication levels would probably be the true solution to this one. I myself use "they" and "their" to avoid unnecessary addition of gender. Just because you feel its ok to "alternate gender" doesn't make it so.

It's a language not a haircut, why not slap a few fashizzle manizzles in there, or do you hate black people? —The preceding unsigned comment was added by 83.70.229.193 (talk • contribs) 12:43, 11 April 2007 (UTC)

It is not just me who feels its OK to alternate gender, it is commonly accepted usage of the english language. As far as the joke being deragotory or sexist: thats reading way too much into it man.

I just adjusted my medication last week. Besides, the voices in my head have impeccable grammar and are well aware of developments in modern usage.

Your virulence in arguing against an established style is astounding. Like you want to go out of your way to make the language sexist...its wierd.

And your one to talk about styles. I notice your using a rather up to date fashion of english:

You are certnyly not vsyng the syme Englysh vsed by Chawcer.

And Providence has not led your Soul to use Georgian English. As all True Christians know is the English bestowed upon us by God.

So whats your deal? Brentt 18:33, 11 April 2007 (UTC)

It's somewhat ironic that you are calling the use of "she" in the example joke an example of something "over the top politically correct", because I can't see the joke as sexist or derogatory without resorting to being overly politically correct! Certainly no mathematician would regard this as a derogatory joke, as it is common enough in mathematical humor to describe specialists of a discipline as being unable to distinguish between objects beyond some fine-ness of structure.

In order for the joke to have sexist connotations, we would have to assume a person hearing the joke would think it attacks topologists and defames women by implying that a typical topologist is a woman. A far more natural reading would be that topologists are comprised of men and women, and that "she" was simply used as a neutral pronoun in order to avoid excessive use of "he". --C S (Talk) 11:43, 15 April 2007 (UTC)

Are any of you females? Despite this article claiming to be one of the 500 most viewed in the field of mathematics, I doubt many of those viewers are female. This is not stereotype, it's simple empirical fact. I wish I could get my wife to take an interest in math, but the fairer sex is generally not inclined towards that subject, although they excel in other subjects I can assure you. For this reason, to my ears (eyes) it reads funny when I encounter "she" in such an article as this one. But otherwise, it's no big deal. My wife and I both use the term "mankind" and we both agree that when doing so we never get a mental picture of the set of all males, but rather, quite naturally we picture all human beings. Neither one of us is offended. —Preceding unsigned comment added by 74.57.167.18 (talk) 00:25, 14 June 2008 (UTC)

Of the six topologists in my postgraduate office, three are women. Make of that what you will. 130.88.123.49 (talk) 10:50, 13 July 2010 (UTC)

L.E.J. Brouwer missing?

Shouldn't L.E.J. Brouwer be mentioned in the history of the field? I am no specialist, but I've always understood that he has made valuable contributions, and and transformed a topic of curious interest into a well-defined discipline. Cheers/JoepVanDelft, 4th of June.

What is algebraic topology?

This article seems to take algebraic topology too broadly in places, and completely misses the modern field of low-dimensional topology. Low-dimensional topology (meaning dimensions three and four, and sometimes two, and including knot theory) is absolutely not part of algebraic topology; the techniques are just completely different. Algebraic topological techniques do work to solve the interesting problems in the theory of manifolds in higher dimensions, but fail in low dimensions. For instance, Perelman's proof of the Poincaré conjecture is analysis, and not at all algebraic topology.

I made an edit to the otherwise good introduction to reflect this, but the article needs some cleanup throughout and should mention the interesting open and recently solved problems in low-dimensional topology. Dylan Thurston 05:06, 17 August 2007 (UTC)

I'm not sure that I agree. Would you say that a knot was a topological object of study? Would you say that a polynomial was an algebraic object of study? If your answer to these two questions is "yes", then how would you describe the Alexander polynomial, the HOMFLY polynomial and the Jones polynomial? They are algebraic constructions used to study toplogical objects, ergo part of algebraic topology. ~~ Dr Dec (Talk) ~~ 13:56, 2 August 2009 (UTC)

I agree that the Jones polynomial, etc, are algebraic objects used to study topology. That does not make them part of algebraic topology, however, which is widely understood by mathematicians to mean the study of ordinary homology and certain variants but explicitly not low-dimensional invariants like the Jones or HOMFLY polynomials. (It does include the Alexander polynomial, which can be defined in terms of the homology of the universal cover.) Really, just look at any textbook on algebraic topology and see what topics it covers. --Dylan Thurston (talk) 10:35, 13 August 2009 (UTC)

Topologists use the phrase "algebraic topology" to include homotopy as well as homology, and to include low-dimensional topology. For example, to pick a book at random off my bookshelf, "A Concise Course in Algebraic Topology" by May has chapters on the van Kampen theorem, covering spaces, graphs, and CW complexes. Rick Norwood (talk) 13:03, 13 August 2009 (UTC)

May's book doesn't support your assertion, IMO. Low-dimensional topology uses some ingredients that belong under the banner of algebraic topology, that's not in doubt. But the "depth" of use is fairly timid -- you quote the basics about fundamental groups, covering spaces, (co)homology, Poincare duality, etc. Low-dimensional topologists also use the h-cobordism theorem to some extent, handle decompositions and morse theory. But do you see applications of the Adams spectral sequence to 3-manifold theory? Do people studying model categories need geometric flows on 3-manifolds, or do they study various metrics on Teichmuller space? This is where the analogy starts to break down. You may want to say that if low-dimensional topology uses the most basic techniques from algebraic topology then it's a part of algebraic topology -- but if you admit that then you'd also have to say parts of electrical engineering, physics, computer graphics, etc, are part of algebraic topology. You could also turn that argument around and then say algebraic topology is a part of low-dimensional topology. The study of high-dimensional manifolds is dominated by things like the h-cobordism theorem and the structure of the tangent bundle of a manifold. But orientable 3-manifolds have trivial tangent bundles, and h-cobordism isn't true in low dimensions -- in particular the main ideas in the proof aren't true, replaced by things like the loop and sphere theorem. That's "why" low-dimensional topology is distinct from algebraic topology -- the basic tools of algebraic topology do not offer enough insight into low-dimensional manifolds. The methods of low-dimensional topology include aspects of algebraic topology but it's also enhanced by analytic/geometric techniques (as Dylan Thurston mentioned) and combinatorial techniques -- see for example the Rubinstein 3-sphere recognition algorithm. Backing off a little: in a sense all distinctions are artificial, as science, natural philosophy, mathematics and astronomy were all at one point more or less synonymous. But go to a generic AMS meeting and you'll find little overlap between an algebraic topology and a low-dimensional topology special session. Rybu (talk) 23:03, 13 August 2009 (UTC)

I just scanned over the article and I haven't been able to find the spots where algebraic topology is taken too broadly. Could you point them out Dylan? Rybu (talk) 23:12, 13 August 2009 (UTC)

Theorems

I have a doubt about one of them.

It reads:

"Every sequence of points in a compact metric space has a convergent subsequence."

Shouldn't it be "every infinite sequence of points"?

If I have the interval [0,3], isn't (1,2,3) a sequence in it? And it doesn't have a convergent subsequence, does it?

Technically yes. Most authors define a sequence to be a function whose domain is N, that is for them all sequences are infinite. Especially when talking about sequences converging, it is understood that they are infinite, since talking about convergence for "finite" sequences makes no sense. Paul August ☎ 00:51, 26 August 2007 (UTC)

If you want to add the adjective "infinite", go for it. Rick Norwood 12:17, 26 August 2007 (UTC)

Request for applications section

Topology has a myriad of applications both in pure and applied mathematics, and I think this article would be greatly enriched by an "applications" section, if anyone feels up to writing it. The current page talks little about any of these applications. Cazort (talk) 15:34, 2 March 2008 (UTC)

Yeah, I agree.. This article could use an applications section. --Farleyknight (talk) 14:41, 23 December 2008 (UTC)

Introduction

I recently edited the intro to try to make it more accessible (reproduced below without the paragraph breaks). The edit was reverted on the grounds that the notion of "close" was incorrect since in Hausdorff spaces any two points can be separated by disjoint open sets. I was attempting (as has been requested on this talk page several times) to add a more accessible introduction to the article, one that provides some intuition to the subject. While the point made is technically true, I keep thinking of the reader who doesn't know what set theory is coming to this page and all they get is Topology begins with a consideration of the nature of space, investigating both its fine structure and its global structure. Topology builds on set theory, considering both sets of points and families of sets. What does the fine structure of space mean? what does global structure mean?

Anyway, just interested in hearing more opinions about the subject. Triathematician (talk) 01:41, 20 March 2008 (UTC)

Topology (Greek topos, "place," and logos, "study") is a branch of mathematics that is an extension of geometry. Topology begins with a consideration of the nature of space, investigating both its fine structure and its global structure. In particular, topology captures the notion of proximity without the need for a notion of distance.

The word topology is used both for the area of study and for a family of sets with certain properties (described below) that are used to define a topological space. The sets in this family are called open sets and, roughly speaking, two points in a topological space are "close" if they are "usually" in the same open set. Of particular importance in the study of topology are functions or maps that are homeomorphisms, which preserve the proximity of points. Informally, these functions can be thought of as those that stretch space without tearing it apart or sticking distinct parts together.

When the discipline was first properly founded, toward the end of the 19th century, it was called geometria situs (Latin geometry of place) and analysis situs (Latin analysis of place). From around 1925 to 1975 it was an important growth area within mathematics.

Topology is a large branch of mathematics that includes many subfields. The most basic division within topology is point-set topology, which investigates such concepts as compactness, connectedness, and countability; algebraic topology, which investigates such concepts as homotopy and homology; and geometric topology, which studies manifolds and their embeddings, including knot theory.

I agree it captures the notion of proximity without using distances.

In this article and most topology books they do indeed use a topology as a set of open sets, but some books in point set topology come from a more set theoretic view and use a topology as a set of families of neighborhoods, and derive the open sets. That aside, I don’t think the statement “’close’ if they are ‘usually’ in the same open set” encompasses all topologies.

The third change seems to me to rely on the second. Homeomorphisms preserve continuity of bijections, and thus preserve open sets.

My personal opinion: use the first of your changes, not the latter two. GromXXVII (talk) 13:03, 20 March 2008 (UTC)

If the idea is to convey the flavor of the subject, I think it is wrong to use either the current version or the new suggestion of discussing "proximity". I think Wikipedia can get a cue from the Brittanica's article on Topology (which was written by RH Bing). It emphasizes the equivalence between objects under natural deformation operations and that topology is concerned with qualitative properties conserved by such deformations. I think to focus on the definition of topological space in the introduction (by basically substituting the word "proximity" for "open neighborhood") detracts from this kind of explanation. ]]

Currently both the introduction and "elementary introduction" section (which appears later) are something of a mess. Probably they can be shortened and combined favorably. --C S (talk) 06:07, 9 May 2008 (UTC)

Ok I cleaned up the lede a bit. It's not perfect, but I think it's certainly a big improvement over what used to be there. Triathemathematician has a point: what does "...investigating both its fine structure and its global structure" mean? I never understood, and if a topologist can't, what is the lay reader supposed to take from it? So I deleted that stuff.

Something I'm considering, which could be controversial, is to remove the list of theorems. I don't think it really help the article to see such a list of theorems, especially when there really is no standard or criteria being applied. Another is to remove the section on the "formal definition". Strange as it may seem, I would argue that going into these set theoretic details really does not give the intelligent layman a good view of the subject. In fact, I wonder if it's really even necessary to give the definition of topological space in the article.

The model I'm considering is geometry, which on the whole, is a lot better written. Note that article doesn't give a list of theorems in Euclidean geometry (which I feel is the analogue to the list on general topology theorems) nor does it get bogged down in details. --C S (talk) 15:44, 20 July 2008 (UTC)

I notice Rick Norwood reinstated the "fine structure and its global structure" bit. As I said, I have no idea what it means, but I'll leave that up to Rick to explain further. Obviously I'm a bit biased, but I think "...topology involves the study of properties that describe how a space is assembled, such as connectedness and orientability..." was much more informative (and beautiful) compared to Rick's "...topology involves the study of properties that describe both the fine structure and global structure of space. Key concepts in topology include compactness, connectedness, and orientability."

Additionally, "Of particular importance in the study of topology are the deformations called homeomorphisms. Informally, these functions can be thought of as those that stretch space without tearing it apart or sticking distinct parts together. A more abstract notion of deformation is homotopy equivalence, which also plays a fundamental role." was changed to "Of particular importance in the study of topology are the functions called homeomorphisms, are continuous with a continuous inverse. Another class of functions that is important in topology are the homotopy equivalences, which stretch space without tearing it apart or sticking distinct parts together." Well, in one obvious sense, homotopy equivalences can and do stick a great many distinct parts together (e.g. any contractible space is homotopy equivalent to a point). I also think if accessibility is a concern, talking about functions and continuous inverses is not the way to go. That can be done at a later point in the article. --C S (talk) 03:15, 13 January 2009 (UTC)

How space is assembled is one area of topology, and orientability and connectedness are examples of global properties. But fine structure properties, such as the Haussdorf property, are also important. Not all topological spaces are manifolds.

C S: Maybe there is no way to save the old saw about "stretching but not cutting". Can you suggest a way? I wonder what non-mathematicians make of the "can't tell the difference between a coffee cup and a donut" maxim?

It is going to be hard to say anything at all beyond the "coffee cup to a donut" level about topology if we can't talk about continuous functions. I'm also concerned about accessibility, but if a person is curious about topology and doesn't have some idea of what a continuous function is, I think they need to learn about continuous functions first.Rick Norwood (talk) 14:19, 13 January 2009 (UTC)

I don't understand your non-sequitor about manifolds. Much of topology has to do with spaces assembled from pieces (like CW complexes) and that is how a lot of spaces are studied. As for "fine structure", you apparently mean that to imply "not global", but your single example of such, Haussdorfness, is a global property.

As far as I can see, mentioning that topology studies global properties of space and also "fine" properties (still don't know what that means) is near meaningless. It's not specific to topology by any means, nor conveys meaning to someone who didn't know anything about topology (or I would wager, even to someone who knew about topology). On the other hand, the wording about studying properties arising from how spaces are assembled (in contrast to metric properties), I believe, does convey something very topological, even to the laymen. It seems to me that the kinds of arguments that rely on how cell structures are patched together are essentially very topological and captures a good chunk (although of course not all) of how topology is done (including the algebraic and geometric topology areas).

I think it's a misconception that a person needs to study continuous functions to learn some topology. It certainly is possible to teach a lot of topology (and I have done so) without going through a rigorous definition of continuous function. Perhaps it's good to keep in mind Brouwer's theorem that every function is continuous. Brouwer was, of course, a constructivist. His point was that as long as you stuck to certain constructive methods of creating new functions, the result was continuous. His theorem is actually a rigorous mathematical result in a certain framework, but my point here is that for the purpose of this article, the functions that people naively define in "geometric" contexts is often continuous. The way the article is currently structured, I don't know if there's a point to including a rigorous definition of continuous. On the other hand, I would not be opposed to such, but I don't believe it needs to take place in the lede section. --C S (talk) 23:52, 13 January 2009 (UTC)

confusing description of the distinctions between homotopy/homeomorphism

the distinctions between the two are quite confusing. for example when it reads that "Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing." one would think that the deformation of the coffee cup into the doughnut is an example of homeomorphism because there is no cutting and no gluing. but as i understand it (and the caption reads) that is not homeomorphic but homotopy. furthermore that same graphic is used in both the homeomorphism and homotopy articles and is identified as an example of each in each article.

furthermore the distinction between the various letters of the alphabet does not make sense under this "intuitive definition." as i understand now (which is admittedly not that great of an understanding)the distinction of letters makes sense under the "intuitive" definition only if you think of the letters as lines and nodes in the mathematical sense as opposed to actually looking at said letters and seeing that they are not really nodes and lines but solid shapes. reading the definition of homeomorphism i would make the categorizations marked as homotopy because for example the top of a T can be pushed into the stalk of the T without any cutting or gluing. furthermore if this is wrong that's seem indicative of the confusion created by the article.

anyway my suggestion is that the graphic be moved so it is closer to what it depicts (if that it is what it depicts) and that the intuitive definition be reworked or removed.Beckeckeck (talk) 10:28, 29 December 2008 (UTC)

edited for clarity Beckeckeck (talk) 10:35, 29 December 2008 (UTC)

Attempt to address C.S.'s objections.

C.S. has raised some valid points, and I've attempted to address them, and also to keep at least the first paragraph readable by a non-mathematician. Help with this rewrite will be appreciated. In particular, I have grave doubts about my example of a homeomorphism: y = x³, which may well do more harm than good. Feel free to delete it. Rick Norwood (talk) 14:44, 13 January 2009 (UTC)

Note to C.S. H and K are not homeomorphic. Can you fix that in your illustration? While you are at it, you might want to rearrange the classes as follows: homeomorphism classes: two holes no tail, one hole no tails, one hole one tail, one hole two tails, no holes one tail, no holes three tails, no holes four tails, H, and K. Homotopy classes: two holes, one hole, no holes. Rick Norwood (talk) 15:07, 13 January 2009 (UTC)

I hope to have time later to make more complete response to your comments, but for now let me comment that H and K are homeomorphic in my illustration. It may help to look close and note they both have two points of valence three. --C S (talk) 23:28, 13 January 2009 (UTC)

Ok, fixed in the article. Rick Norwood (talk) 13:47, 14 January 2009 (UTC)

Some minor points

I just made a couple of minor edits - I changed "The branch of mathematics now called topology" to just "Topology" in the History section, as the longer formulation seemed pointlessly verbal.

I also changed "All sets in T are called open" to "The open sets in X are defined to be the members of T" to clarify that this is actually a definition of open in this context (and, for example, rule out the possibility that a set not in T might be called open). I'm still not happy with this wording, however - perhaps for clarity and consistency the phrasing of Topological space should be used - "The sets in T are the open sets, and their complements in X are called closed sets."

Finally, reading the first few paragraphs, nowhere is it actually explained what Topology is about:

"Topology (Greek Τοπολογία, from τόπος, “place”, and λόγος, “study”) is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others."

Mentions some related areas and its development, but fails to give a quick explanation of what the term currently means, which belongs in the first sentence. --Joth (talk) 21:19, 28 May 2009 (UTC)

Does that word actually exist in Greek? Or, more precisely, did it exist before modern mathematicians invented it in other languages? Peter jackson (talk) 15:00, 2 June 2009 (UTC)

[Merriam-Webster http://www.merriam-webster.com/dictionary/topology] gives a date of 1850 for the word.

Joth does have a point about the intro. You do need to get down to the Elementary introduction section before there is anything which actually describes the subject

the study of qualitative properties of certain objects (called topological spaces) that are invariant under certain kind of transformations (called continuous maps), especially those properties that are invariant under a certain kind of equivalence (called homeomorphism).

we do need something better in the lead. --Salix (talk): 18:10, 2 June 2009 (UTC)

How about we change the first line to the following

Topology (Greek Τοπολογία, from τόπος, “place”, and λόγος, “study”) is a major area of mathematics concerned with spatial properties that are preserved under bicontinuous deformation; that is, stretching without either tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others.

With help from http://en.wikipedia.org/w/index.php?title=Topology&oldid=75290539. Perhaps the phrase 'bicontinuous deformation' still sounds a bit too technical for the first sentence, but I can't think of a better way to phrase it. Any ideas? --Joth (talk) 07:03, 5 June 2009 (UTC)

Ernest Klein, A Comprehensive Etymological Dictionary of the English Language, Elsevier, 1971, page 772, says it's from topos & logia. No mention of any supposed Greek word topologia. Peter jackson (talk) 15:13, 5 June 2009 (UTC)

Perhaps we should just say "from the Greek τόπος, “place”, and λόγος, “study”)" instead then? --Joth (talk) 21:39, 5 June 2009 (UTC)

Since it's good to be bold, and nobody's had any problems with them, I've made both of the changes suggested above. Any problems, please comment. --Joth (talk) 08:43, 6 June 2009 (UTC)

I like the new first paragraph. I agree that the term "bicontinuous" is too technical. It would leave the reader wondering about what it means, even with a link. I think it's better to avoid that term completely, especially since the "tearing or gluing" portion of the sentence is only really true for certain instances of topologies. Perhaps we could say "concerned with spatial properties that are preserved under deformations that involve stretching but no tearing or gluing." Triathematician (talk) 12:04, 25 June 2009 (UTC)

I like your suggested change, so I've made it, with one small change - I put in 'deformations of objects', since I wonder whether we should say what is actually being deformed here. Objects is still the wrong word, but surely it's more than just shapes... -- Joth (talk) 11:15, 27 June 2009 (UTC)

Quotient space confusion

Sorry for posting here the talk at quotient space doesn't seem to be viewed often.

If a quotient is injective it's a homeomorphism right? So why can't we just say two spaces are "quotient" equivalent if there exist a quotient map between them? Instead of constructing the partitioning of the domain space X and then showing there's homeomorphism between the partition space and the range... So what I mean is that just like homeomorphism can we think of the points in X substituted into the range space Y except it doesn't have to be injective and in a way that it's done continuously.Standard Oil (talk) 21:10, 17 July 2009 (UTC)

1st question, true. 2nd question, I don't see the point. For example, your notion of "quotient equivalence" is not an equivalence relation, since a 1-point space is a quotient of a 2-point space, but the reverse is false. So I don't understand the point of your 2nd question, and a lot of what you say seems pretty vague. Rybu (talk) 07:48, 14 August 2009 (UTC)

Standard Oil: If a quotient map is one-to-one, then it is a homeomorphism, true. And every homeomorphism can be viewed as a one-to-one quotient map. But since "homeomorphism" and "one-to-one quotient map" mean the same thing, the more common word, "homeomorphism", is invariably used. All of the interesting examples of quotient maps are not on-to-one. A good source for further reading on this subject is Munkres, Topology, 2nd edition, section 22. Rick Norwood (talk) 12:56, 14 August 2009 (UTC)

Yea I know what I said was pretty vague... Never thought that it's not an equivalence relation, that answers my question. It's about this theorem: Let g: X->Z be a quotient map. Let X* be the following collection of subsets of X: X*={(g^-1)({z})| z belongs to Z}. Give X* the quotient topology. Then the map g induces a homeomorphism f: X*->Z. I was wondering why bother partitioning X to X* at all. Meh this was quite a while ago no one answered me and what do you know I was getting on to check something about Richard's paradox and dropped by to see someone replied. Sorry for my wikipedia skillsStandard Oil (talk) 14:08, 14 August 2009 (UTC)

Ah, IMO the main reason for that construction is to prove a slightly different proposition -- given any continuous function f: X --> Y you can factor it as a composite of a quotient map followed by an injective continuous function. Rybu (talk) 03:39, 16 August 2009 (UTC)

Supposed Latin

The word "geometria" is said to be Latin, but it seems to be Greek. —Preceding unsigned comment added by 81.148.78.84 (talk) 16:27, 29 September 2009 (UTC)

It's both (the Latin word being borrowed from Greek). But the article is talking about the phrase geometria situs, which is Latin, not Greek. --Zundark (talk) 16:36, 29 September 2009 (UTC)

Coffee cup to doughnut image

Would it be better to have the 'coffee cup to doughnut and back' image right at the start of the article (instead of the Mobius strip) ? It would be more eyecatching and illustrates the popular idea of topology (I know, there is more to topology than this, but it certainly grabs the attention more than a static Mobius strip. Thoughts on this plz ? Thx. MP ^{(talk•contribs)} 16:17, 24 October 2009 (UTC)

Images in articles should be explanatory, not eye-catching. That said, demonstrations of topological equivalences betray more information about the structure of the field than does an example of a thing that has a topology, because in only one is it at all apparent what part of the thing is the topology. LokiClock (talk) 09:34, 23 November 2009 (UTC)

So, what is your opinion on including the coffee cup to doughnut image at the start ? Yes or no. MP ^{(talk•contribs)} 21:29, 23 November 2009 (UTC)

Hairy Ball Theorem not self-explanatory

"Similarly, the hairy ball theorem of algebraic topology says that 'one cannot comb the hair flat on a hairy ball without creating a cowlick.' This fact is immediately convincing to most people"

How is this immediately convincing? Sorry, but I was not raised with the necessary understanding of the inner workings of combing hairy balls and creating cowlicks. In fact, I have barely any idea what this is supposed to mean in a literal sense, much less how to put it into a mathematical context. The second statement just deflects the dire necessity to explain the hairy ball theorem, its meaning, its context, and its significance. If this is really so obvious after having seen someone combing a hairy ball, an animation or video of this happening should be supplied. LokiClock (talk) 09:31, 23 November 2009 (UTC)

Literally, if a ball is covered in hair, it is impossible to comb the hair so that it is flat against the ball at every point. Sławomir Biały (talk) 14:30, 24 November 2009 (UTC)

Suggested Edits to Topology Page

My e-mail is Tim-J.Swan To anyone who reads this: I cannot currently edit the image I intend to, however, I believe "X" should be removed from its homeomorphism category on the Topology page and placed among K and H. The "bar" that they contain I don't believe should be considered an element in homeomorphism. In my mathematical observation and after double checking the definition of homeomorphism, I conclude that just because the text has another angle in it, it still is part of the node connecting 4 endpoints. If the image can be fixed and the text to match it, I would appreciate it. Feedback would be nice. —Preceding unsigned comment added by Tim-J.Swan (talk • contribs) 15:12, 9 January 2010 (UTC)

No. How would you define a function going from the X to a H for the homeomorphism? Dmcq (talk) 11:10, 13 July 2010 (UTC)

Tone

"This is harder to describe without getting technical" does not seem like a proper tone for this article --187.40.180.230 (talk) 16:27, 24 November 2010 (UTC)

grammar in mathematical def

What's it mean, "arbitrarily many elements", and "finitely many elements" ? --Jerome Potts (talk) 20:26, 29 November 2010 (UTC)

Equivalence Classes Images (Example)

Minor suggestion, the examples (images Alphabet_homotopy.png, Alphabet_homeo.png) are not obviously clear. Reason 1: the images are side by side with no clear separation - suggest separating them vertically with explanation paragraph between. Reason 2: Each set is an individual example, but they are lumped together in the formatting of the images - suggest listing them vertically or in a grid. While the set notation is obvious to the initiated it took me a minute to understand these 2 images. Please reply if there are better ways of doing things - I can also create revised images. Littledman (talk) 00:49, 30 May 2011 (UTC)

The first sentence

2011-8-17. In my opinion, the first sentence of the topology article should be this:

"Topology is the study of continuity and connectivity."

Today I have added this sentence in the "Elementary introduction" section. I disagree with the statement that topology is the study of properties which are invariant under continuous transformations because continuous functions are defined to be those functions which preserve the topology. It's clearly a circular definition. It's also incorrect because half of continuity is concerned with functions, not properties of sets. Continuity of functions is the principal focus of the applications of topology to analysis, for example, topological vector spaces and partial differential equations existence/uniqueness/regularity theory.

The word "continuity" covers those topological questions which relate to functions. The word "connectivity" covers the topological questions about objects, which is half of topology which is easy to introduce to non-mathematicians. The function-continuity half of topology is largely inexplicable to non-mathematicians, being mostly technical and difficult to motivate. Thus the "continuity and connectivity" definition for topology covers both the analysis and geometry halves of topology.
Alan U. Kennington (talk) 06:16, 17 August 2011 (UTC)

First point: You seem to be claiming a circularity in a statement like (paraphrasing for simplicity): Topology is the study of functions that preserve the topology. But that's not circular at all. The two instances of the word topology refer to completely different things. The first instance is a mathematical discipline; the second is a mathematical object. Totally different. --Trovatore (talk) 08:38, 17 August 2011 (UTC)

I'm fully aware of the difference between "topology" (uncountable noun) and "a topology" (countable noun). That is not relevant to what I was saying. Anyway, the circular folk definition is given in large numbers of introductory books and pop science articles. So it's appropriate to give it here also. The more important issue is how to correctly characterize the subject of topology in one snappy sentence. Topology in analysis is certainly not about the properties which are left invariant by continuous functions. That would be a Felix Klein style of "definition". Topology in analysis is concerned with the continuity of maps, operators, functions and transformations, and sets tend to have multiple topologies in the same context. Thus the focus is typically on the relations between multiple topologies on the same set and on the continuity of functions with respect to various topologies for domain and range. The interest is not so much on invariant properties of sets under homeomorphisms. That's the connectivity branch of topology.
Alan U. Kennington (talk) 09:26, 17 August 2011 (UTC)

If it's not relevant to what you were saying, then exactly what is the circularity? It doesn't have anything to do with defining topology in terms of itself; you seem already to have agreed to that. --Trovatore (talk) 09:32, 17 August 2011 (UTC)

I should say that I don't entirely disagree with you that the definition is too glib. A better approach would be not to try to define topology at all, but just to identify it, by talking about some of the issues with which it is concerned. See my most recent remarks at talk:mathematics, which are quite analogous.

However, if by "connectivity" you're getting at topology-in-the-computer-science-sense, I sharply disagree that this article should talk about that at all. That's almost but not quite entirely unlike topology in the mathematical sense. I would go so far as to call it a "misnomer", except that of course computer scientists are entitled to come up with their own jargon as they see fit. But it doesn't belong in the same article as mathematical topology; it just doesn't have enough points of commonality with it. --Trovatore (talk) 09:46, 17 August 2011 (UTC)

I'm very aware of the computer science (and communications network) meaning of "topology", and it is in fact related to the mathematical analysis definition of topology, although not directly enough to justify being on the same page as the mathematical definition. (I cover discrete/network topology in a section of the topology chapter in my differential geometry book, for example.)

What I had in mind was the analytical-topological concepts of connectedness, multiple connectedness and the full range of algebraic topology tools such as homotopy, homology, simplicial complexes etc. All of the business about donuts and teapots is related to the classification problem, and such classification proceeds by examining the connectivity properties of sets or manifolds or whatever. There is a huge corpus of topology related to this classification issue in finite dimensional manifolds. But there is also a whole corpus of topology related to topological vector spaces, where the connectivity classes are all essentially trivial. Every topological vector space is connected and simply connected. There are no connectivity classes to classify. Here the issues relate to continuity of maps and functions with respect to varying strengths of topologies. The topology classes in this kind of topological linear space situation are more related to local compactness, local convexity, separability and related properties.

This is why I split topology into the continuity branch, where the focus is on the functions and multiple-choice topologies, and the connectivity branch of topology where the focus is on classifying manifolds into connectivity equivalence classes. I claim that my 8-word proposed first sentence communicates to the non-specialist the dual foci of topology — smoothness (i.e. continuity) of functions and connectivity of sets.
Alan U. Kennington (talk) 10:53, 17 August 2011 (UTC)

Ah, so by "connectivity" you're talking about stuff like, in the simplest case, genus, right? Homotopy and homology groups and on from there?

If that's what you're getting at, I'm afraid I do not think that "connectivity" communicates that to the non-specialist. The non-specialist is likely to see "connectivity" and think you're talking about which machines can talk to which other machines via what networks, especially if he/she has seen that meaning somewhere as used by computer scientists. --Trovatore (talk) 18:41, 17 August 2011 (UTC)

How about "connectedness"? I think it's better than "genus". Homotopy and homology groups are tools for determining how a topological space is connected. The simplest concepts are "connected", "simply connected" and "multiply connected". I don't see how "connectedness" is a confusing word. If a set has two components, it's disconnected. That's intuitively clear to the non-mathematician. If a set in real 2-space has a hole, it's doubly connected. Once again, this is something which the non-mathematician can easily understand. Thus:

"Topology is the study of continuity and connectedness."

I think that's a lot more meaningful — and correct — than this:

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation.

The analysis branch of topology — like topological vector spaces and distribution theory and functional analysis — is not concerned with invariants of subsets of topological spaces under homeomorphisms. So the first paragraph of the article is simply wrong. It only describes the connectedness (topological genus) branch of topology, usually in low-dimensional manifolds. That's just one amongst many applications areas. In my work, all topological spaces are linear spaces, for which the first paragraph of the article has no relevance. Therefore it is important to mention that continuity itself is a major subject of study. The first paragraph also does not mention analysis as a primary source of topological concepts. It's not just geometry and set theory. Anything to do with limits is a topological matter. So that includes all of analysis. Homeomorphism-invariant properties can be illustrated in diagrams which the non-mathematician can more or less understand. However, barrelled spaces, semi-norms, Banach spaces, Hilbert spaces, etc., are enormously important in analysis. These are excluded from the current first paragraph. But if it is mentioned that continuity is a major topic of topology, then the analysis branch of topology would be included.
Alan U. Kennington (talk) 07:56, 18 August 2011 (UTC)

Yes connectedness sounds good, the current bit about continuity just is about local properties rather then global properties. I think perhaps it would be better to have two sentences though and after the basic sentence about continuity go on to say about a couple of simple global properties like connectedness that people without experience of the subject can relate to. Dmcq (talk) 08:18, 18 August 2011 (UTC)

That's a very good point which I hadn't thought of. Yes, the analysis applications of topology are almost all local, whereas the classification of finite-dimensional manifolds into equivalence classes using homeomorphism invariants is a global matter. Connectedness is a global issue whereas continuity is a local issue. The most important thing, though, is that someone has marked this article as "top priority" and "vital", but "Start-Class" or "Class C", which doesn't sound very good to me. Since the article gets 1330 views per day, I think that at least the first paragraph should be meaningful to most readers, and preferably as accurate as possible. If the reader is flummoxed by the first paragraph, they are unlikely to proceed further. I don't want to edit the first paragraph unilaterally and get flamed for it. So I'm hoping that we can decide on a better paragraph which everyone can live with!
Alan U. Kennington (talk) 08:57, 18 August 2011 (UTC)

Connectedness works much better than connectivity, I think, and maybe this is an important point to get across. Algebraic topology is something I know very little about, so I might not have thought of it — if you'd just said topology is the study of continuity, I would not really have noticed that anything was missing. (In descriptive set theory the topologies we're usually interested in are those of zero-dimensional Polish spaces, where there is no connectedness to worry about, and that's kind of why they're used.)

So I kind of like topology is the study of continuity and connectedness, but how are you going to source it? Even if you can, is it possible that it's leaving something out? --Trovatore (talk) 17:38, 18 August 2011 (UTC)

When you say source it, I guess you mean to find it written in a published work to quote. But my own definition arises from dissatisfaction with the published single-sentence definitions of topology, which are usually included in very elementary presentations for the general public, e.g. in popular science magazines. My own simple definition appears in my own book draft, which is not yet published. So I guess that doesn't count! So here is something from the 1963 book "Introduction to topology and modern analysis" by G.F. Simmons (Preface, page viii) which was my textbook when I first learned serious general topology.

Historically speaking, topology has followed two principal lines of development. In homology theory, dimension theory, and the study of manifolds, the basic motivation appears to have come from geometry. In these fields, topological spaces are looked upon as generalized geometric configurations, and the emphasis is placed on the structure of the spaces themselves. In the other direction, the main stimulus has been analysis. Continuous functions are the chief objects of interest here, and topological spaces are regarded primarily as carriers of such functions and as domains over which they can be integrated. These ideas lead naturally into the theory of Banach and Hilbert spaces and Banach algebras, the modern theory of integration, and abstract harmonic analysis on locally compact groups.

This is surprisingly close to what I was saying, namely that topology splits into those two branches of investigation. The first focuses on the structure of the spaces (which I claim is related to homeomorphism invariants such as homotopy as tools to classify spaces into homeomorphism equivalence classes). The second focuses on the functions (which I claim is related to "continuity").

I'm not sure that everything in wikipedia has to be a quote from somewhere. When other encyclopedias have been written in the past, experts in each field were expected to come up with their own very short and comprehensible definitions which would be helpful to non-specialists. Most of the serious academic books on topology will define it in terms which assume a few years of university-level mathematics, whereas wikipedia should be maximally comprehensible to anyone. So we need to use a kind of non-technical language which would not appear on page 1 of an academic topology textbook. I think that my above quote from a 3rd year university textbook does support the split of topology into two branches which I claim. My most important claim is that we should not forget the "continuous functions" half of topology when giving the first one-sentence definition. My secondary claim is that the "generalized geometric configurations" half of topology is best summarised as the study of "connectedness (of sets)", or something like that.
Alan U. Kennington (talk) 18:27, 18 August 2011 (UTC)

I'd like to suggest keeping the introduction more or less in the same tone as it currently is. I'm not so sure I like emphasizing a "split" right at the beginning of the article. It's distracting. The article roughly describes the early motivation for topology -- "analysis situs". This came up in the considerations of people like Gauss, Riemann and Poincare. The point-set aspects of topology that Alan refers to in the context of analysis, you might call the "soft analysis" aspect of topology. This was a slower thing to develop, and in a sense was more of a 2nd order phenomena in that it was not part of the core, early motivation of topology. Global properties of ODEs, fixed points, complex surfaces, knots, local properties of algebraic varieties, Poincare duality and such, these were more of the driving force. So I think it's important to keep this geometric character in focus when telling people what the subject is about. Rybu (talk) 07:54, 19 August 2011 (UTC)

I certainly don't agree with shorting general topology! To me, that's first and foremost what topology means. However it had never occurred to me that the "properties preserved under homeomorphism" formulation wasn't about general topology. It seems to me that you reach that interpretation only if you insist on the homeomorphisms being between entire spaces, rather than neighborhoods, which is the way I tend to think of it. --Trovatore (talk) 08:05, 19 August 2011 (UTC)

The analysis half of topology started with the ancient Greeks. They used the "method of exhaustion" to take limits of areas and volumes. Newton and Leibniz built the core of modern mathematics by bringing limits to the foreground of mathematics. Everything to do with limits is topology — the analysis half of topology. It is analysis which has been the bedrock of physics for 350 years now. Analysis is about limits, and limits are probably the most fundamental concept of topology. The analysis side of topology includes limits, convergence, metric spaces, compactness, etc. Fourier series, Fourier transforms and Laplace transforms use sophisticated topology concepts, as does also the calculus of variations. Then distribution theory relies heavily on semi-normed spaces. The majority of mathematical physics uses Hilbert spaces, Banach spaces, distribution spaces, variational methods etc. These rely heavily on an understanding of the topological structure on these spaces.

So I would say that the analysis half of topology is the most ancient and most significant aspect of all of mathematics. The word "topology" was coined only about 160 years ago, as we all know, by Listing. I have downloaded and read through the book by Listing, and I can tell you that it mostly not about what we would call topology today. It is mostly about some recreations in combinatorics. The term "analysis situs" really gives the game away, I think. Topology is a part of the subject of analysis. Since that is my own focus, I am very familiar with the depth of history of the analysis half of topology. The classification of topologies into homeomorphism equivalence classes using homoemorphism-invariant structures and properties is only a topic in the global analysis of finite-dimensional manifolds, a subject of quite recent interest. Most of the "pictures" of topology in introductory books come from this geometrical side of topology — because you can't make useful pictures of the function-continuity half of topology. But just because there are no "pictures", that doesn't mean that the analysis half should be hidden. I think that at least somewhere on the Topology wikipedia web page, there should be a full section on the truly important half of topology — the analysis which is the basis of most of mathematical physics and engineering. The homeomorphism-invariants side of topology is really mostly of a recreational character — pure mathematics.
Alan U. Kennington (talk) 08:32, 19 August 2011 (UTC)

Alan, it seems like you're changing the nature of the discussion. You started this discussion concerned with the 1st sentence of the article, and now you're talking about "somewhere in the article". The fact that you call the aspect of topology you care about the "truly important half" and the other half "recreational" seems pretty out-of-touch with the subject. I'd like to suggest giving it a break for a while. Rybu (talk) 22:05, 19 August 2011 (UTC)

Rybu, I wasn't changing the nature of the discussion. I was trying to find a compromise whereby topology would be defined correctly at least somewhere in the article. I'm in touch with both halves of the subject, believe it or not. But your ad-hominem argument shows that there is not point in further discussion. Criticising the person's competence is not a valid argument, as far as I know. Everyone thinks their own tree is the whole universe. So I'm leaving this tree now. I'll remove this page from my watch list. Enjoy!

Alan U. Kennington (talk) 01:16, 20 August 2011 (UTC)

Science News resources, regarding Euler ... Topology tackles Königsberg and the entire universe

Mathematicians think of everything as rubber October 22nd, 2011; Vol.180 #9 ... from the 1940 Archive to now 97.87.29.188 (talk) 23:20, 1 November 2011 (UTC)

See Seven Bridges of Königsberg. 99.181.138.228 (talk) 03:43, 2 November 2011 (UTC)

Interesting. Well, the [[]] is not Topology, but Graph theory. Do we need to split off an article Topology (media) to handle all the misconceptions of topology?

It is true that the media, even good sources such as Science News, often get math wrong. But note that to make the Seven Bridges of Königsberg into a graph theory problem, you need to prove that the underlying graph is a strong deformation retraction of the (surface of) the bridges, which is topology. Graph theory still has strong connections to topology, see AMS Subject classification 57M15. Rick Norwood (talk) 13:27, 2 November 2011 (UTC)

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