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

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.
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 $A=\{x_{n}:n=1,2,\ldots \}$ 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, T) and (X, T*) are topological spaces with T* finer than T and (X, T*) is limit point compact, then so is (X, T).

Notes

^ 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

James Munkres (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.
Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).
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

See also

Notes

References