Wikipedia:Reference desk/Archives/Mathematics/2006 October 11

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Topological group counterexample[edit]

This may be a little advanced for this page, but I'll give it a shot anyway. Does anyone know of an example of a topological group which is connected but not path-connected? Or do such groups not exist? Of course, such a group cannot be locally path connected and so cannot be anything nice like a Lie group. -- Fropuff 17:59, 11 October 2006 (UTC)[reply]

Consider in RxR, with the usual topology, X={(x,sin(1/x)) | x is in [1,0)}, and Y={(0,y) | y in [-1.1]}, now, XUY is connected and not path-connected. By the way, it is not an advanced topic.

This example is a topological space which is connected but not path-wise connected (if [1,0) is interpreted to mean (0,1]). However, the question concerned a topological group; or, in other words, a set which simultaneously has a group structure and a topology, such that the group actions 'respect' the topology in a natural manner. That is indeed a more advanced topic, and a harder question to answer. JoergenB 21:08, 11 October 2006 (UTC)[reply]

Right :P, sorry my wrong

But there still remains the interesting question as to whether a topological group which is connected has to be path-connected. I can't believe that the topologists sine curve could be a topological group - but I can't see any obvious reason why it is impossible to define a group structure on the points of the curve ... Madmath789 21:47, 11 October 2006 (UTC)[reply]

It's not possible. A topological group has to be homogeneous to a certain extent; given any two points you can define a (topological) autohomeomorphism of the whole space that takes one of the points to the other one. (Just multiply everything by the quotient of the two points.) The space above does not have this property; there's a point that has no path-connected neighborhood, whereas every other point does have a path-connected neighborhood. --Trovatore 21:51, 11 October 2006 (UTC)[reply]

Spot on! I think I should have known that ... Madmath789 22:06, 11 October 2006 (UTC)[reply]

I do not believe such groups exist. Google doesn't seem to be coming up with either a proof or a counterexample, so I'd try Bourbaki next.

RandomP 18:47, 11 October 2006 (UTC)[reply]

Morava states here that the Pontryagin dual of

\mathbb {Q}

should be a counter-example.--gwaihir 21:14, 11 October 2006 (UTC)[reply]

Thanks for the suggestion. I've never seen Pontryagin duality defined for non-locally compact groups. I assume the definition is the same. I'll have to try and think about why the dual of Q is not path-connected (if that is indeed the case; Morava doesn't sound entirely sure). -- Fropuff 05:10, 12 October 2006 (UTC)[reply]

Hopefully this is a less advanced question--What nontriv. spaces are everywhere connected but nowhere pathconnected? I'm trying to piggyback on Trovatore's reasoning above.Rich 18:57, 12 October 2006 (UTC)[reply]

Okay, I have a nice answer compliments of Daniel Asimov (who responded to a question posted on sci.math.research by Rich). The p-adic solenoid is a compact, connected abelian topological group which is neither path-connected nor locally-connected. I have yet to work out what the path component of the identity is, or the group of path components, but at least I have something to play with. -- Fropuff 06:36, 18 October 2006 (UTC)[reply]

TeX Supporting non-Wikimedia sites/servers for editing/previewing?[edit]

Given the uselessly slow loading of images (including TeX) the last few days (at least sporadically), it seems practical to find other wiki sites that are fully TeX supported (and with open preview privileges) to work on articles (some, like http://uncyclopedia.org, seem to support most TeX, but not the most recent commands/effects——e.g., "{\color{white}.}V=Q", to add a hidden character for space value at the beginning). Anyone know of any alternate sites to use for such purposes? ~Kaimbridge~19:14, 11 October 2006 (UTC)[reply]

Be grateful if you could be more specific than http://uncyclopedia.org where I could not find anything related to TEX. Thank you in advance. Twma 01:06, 12 October 2006 (UTC)[reply]

Just hit the "Random page" link, then the "edit": Blank out the text that is there, cut and paste whatever you are working on, work on it, then cut it back out and save it wherever you had it before (you are just "previewing" on uncyclopedia, not saving anything).

Add a space and a negative space if you want to force TeX. \colour (IIRC) is not TeX, it most probably is a LaTeX addition, and it's not the best solution anyway. Dysprosia 02:07, 12 October 2006 (UTC)[reply]

After hitting the "Random page" link on the left column menu, the first time, I got a cooking page. For the second time, I got Editing Top Ten Numbers, etc. I do not understand what is a nagative space. I typed in space and delete. I typed in \frac{1}{2} but it did not show the fraction. Tried again with $\frac{1}{2}$ but failed again. I am interested in HOW IT WORKS but not whether it works. Clearly, it was not working probably due to my misunderstanding of NEGATIVE SPACE.Twma 12:20, 13 October 2006 (UTC)[reply]

Okay, after you landed on the "cooking page", you should have hit the "edit" link at the top, deleted the text, then work on whatever. The wiki TeX endmarkers are "<math>" and "</math>" (not "$"s) and the negative space is "\!", so a small space ("\,") and a negative space cancel out: Using your example, the proper way to format a fraction is "<math>V=\frac{N}{D}\,\!</math>", which should come out as $V={\frac {N}{D}}\,\!$ .
There is a good "cheat sheet" page, Help:Formula (though I prefer an older version, as the format layout is better——besides using the "printable yes" link). P=) ~Kaimbridge~14:13, 13 October 2006 (UTC)[reply]