Limit point compact

In mathematics, a topological space $X$ is said to be limit point compact^[1]^[2] or weakly countably compact^[3] if every infinite subset of $X$ has a limit point in $X.$ This property generalizes a property of compact spaces. In a metric space, limit point compactness, compactness, and sequential compactness are all equivalent. For general topological spaces, however, these three notions of compactness are not equivalent.

Properties and examples[edit]

In a topological space, subsets without limit point are exactly those that are closed and discrete in the subspace topology. So a space is limit point compact if and only if all its closed discrete subsets are finite.
A space $X$ is not limit point compact if and only if it has an infinite closed discrete subspace. Since any subset of a closed discrete subset of $X$ is itself closed in $X$ and discrete, this is equivalent to require that $X$ has a countably infinite closed discrete subspace.
Some examples of spaces that are not limit point compact: (1) The set $\mathbb {R}$ of all real numbers with its usual topology, since the integers are an infinite set but do not have a limit point in $\mathbb {R}$ ; (2) an infinite set with the discrete topology; (3) the countable complement topology on an uncountable set.
Every countably compact space (and hence every compact space) is limit point compact.
For T₁ spaces, limit point compactness is equivalent to countable compactness.
An example of limit point compact space that is not countably compact is obtained by "doubling the integers", namely, taking the product $X=\mathbb {Z} \times Y$ where $\mathbb {Z}$ is the set of all integers with the discrete topology and $Y=\{0,1\}$ has the indiscrete topology. The space $X$ is homeomorphic to the odd-even topology.^[4] This space is not T₀. It is limit point compact because every nonempty subset has a limit point.
An example of T₀ space that is limit point compact and not countably compact is $X=\mathbb {R} ,$ the set of all real numbers, with the right order topology, i.e., the topology generated by all intervals $(x,\infty ).$ ^[5] The space is limit point compact because given any point $a\in X,$ every $x<a$ is a limit point of $\{a\}.$
For metrizable spaces, compactness, countable compactness, limit point compactness, and sequential compactness are all equivalent.
Closed subspaces of a limit point compact space are limit point compact.
The continuous image of a limit point compact space need not be limit point compact. For example, if $X=\mathbb {Z} \times Y$ with $\mathbb {Z}$ discrete and $Y$ indiscrete as in the example above, the map $f=\pi _{\mathbb {Z} }$ given by projection onto the first coordinate is continuous, but $f(X)=\mathbb {Z}$ is not limit point compact.
A limit point compact space need not be pseudocompact. An example is given by the same $X=\mathbb {Z} \times Y$ with $Y$ indiscrete two-point space and the map $f=\pi _{\mathbb {Z} },$ whose image is not bounded in $\mathbb {R} .$
A pseudocompact space need not be limit point compact. An example is given by an uncountable set with the cocountable topology.
Every normal pseudocompact space is limit point compact.^[6]
Proof: Suppose $X$ is a normal space that is not limit point compact. There exists a countably infinite closed discrete subset $A=\{x_{1},x_{2},x_{3},\ldots \}$ of $X.$ By the Tietze extension theorem the continuous function $f$ on $A$ defined by $f(x_{n})=n$ can be extended to an (unbounded) real-valued continuous function on all of $X.$ So $X$ is not pseudocompact.
Limit point compact spaces have countable extent.
If $(X,\tau )$ and $(X,\sigma )$ are topological spaces with $\sigma$ finer than $\tau$ and $(X,\sigma )$ is limit point compact, then so is $(X,\tau ).$

Notes[edit]

^ The terminology "limit point compact" appears in a topology textbook by James Munkres where he says that historically such spaces had been called just "compact" and what we now call compact spaces were called "bicompact". There was then a shift in terminology with bicompact spaces being called just "compact" and no generally accepted name for the first concept, some calling it "Fréchet compactness", others the "Bolzano-Weierstrass property". He says he invented the term "limit point compact" to have something at least descriptive of the property. Munkres, p. 178-179.
^ Steen & Seebach, p. 19
^ Steen & Seebach, p. 19
^ Steen & Seebach, Example 6
^ Steen & Seebach, Example 50
^ Steen & Seebach, p. 20. What they call "normal" is T₄ in wikipedia's terminology, but it's essentially the same proof as here.

References[edit]

Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260.
Steen, Lynn Arthur; Seebach, J. Arthur (1995) [First published 1978 by Springer-Verlag, New York]. Counterexamples in topology. New York: Dover Publications. ISBN 0-486-68735-X. OCLC 32311847.
This article incorporates material from Weakly countably compact on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

[1] The terminology "limit point compact" appears in a topology textbook by James Munkres where he says that historically such spaces had been called just "compact" and what we now call compact spaces were called "bicompact". There was then a shift in terminology with bicompact spaces being called just "compact" and no generally accepted name for the first concept, some calling it "Fréchet compactness", others the "Bolzano-Weierstrass property". He says he invented the term "limit point compact" to have something at least descriptive of the property. Munkres, p. 178-179.

[2] Steen & Seebach, p. 19

[3] Steen & Seebach, p. 19

[4] Steen & Seebach, Example 6

[5] Steen & Seebach, Example 50

[6] Steen & Seebach, p. 20. What they call "normal" is T₄ in wikipedia's terminology, but it's essentially the same proof as here.

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Properties and examples[edit]

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