Countable Compactness

In mathematics a topological space is called countably compact if every countable open cover has a finite subcover.

Equivalent definitions

A topological space ''X'' is called countably compact if it satisfies any of the following equivalent conditions:

:(1) Every countable open cover of ''X'' has a finite subcover.
:(2) Every infinite ''set'' ''A'' in ''X'' has an ω-accumulation point in ''X''.
:(3) Every ''sequence'' in ''X'' has an accumulation point in ''X''.
:(4) Every countable family of closed subsets of ''X'' with an empty intersection has a finite subfamily with an empty intersection.

(1) $\Rightarrow$ (2): Suppose (1) holds and ''A'' is an infinite subset of ''X'' without $\omega$-accumulation point. By taking a subset of ''A'' if necessary, we can assume that ''A'' is countable.
Every $x\in X$ has an open neighbourhood $O_x$ such that $O_x\cap A$ is finite (possibly empty), since ''x'' is ''not'' an ω-accumulation point. For every finite subset ''F'' of ''A'' define $O_F = \cup\$. Every $O_x$ is a subset of one of the $O_F$, so the $O_F$ cover ''X''. Since there are countably many of them, the $O_F$ form a countable open cover of ''X''. But every $O_F$ intersect ''A'' in a finite subset (namely ''F''), so finitely many of them cannot cover ''A'', let alone ''X''. This contradiction proves (2).

(2) $\Rightarrow$ (3): Suppose (2) holds, and let $(x_n)_n$ be a sequence in ''X''. If the sequence has a value ''x'' that occurs infinitely many times, that value is an accumulation point of the sequence. Otherwise, every value in the sequence occurs only finitely many times and the set $A=\$ is infinite and so has an ω-accumulation point ''x''. That ''x'' is then an accumulation point of the sequence, as is easily checked.

(3) $\Rightarrow$ (1): Suppose (3) holds and $\$ is a countable open cover without a finite subcover. Then for each $n$ we can choose a point $x_n\in X$ that is ''not'' in $\cup_^n O_i$. The sequence $(x_n)_n$ has an accumulation point ''x'' and that ''x'' is in some $O_k$. But then $O_k$ is a neighborhood of ''x'' that does not contain any of the $x_n$ with $n>k$, so ''x'' is not an accumulation point of the sequence after all. This contradiction proves (1).

(4) $\Leftrightarrow$ (1): Conditions (1) and (4) are easily seen to be equivalent by taking complements.

Examples

*The first uncountable ordinal (with the order topology) is an example of a countably compact space that is not compact.

Properties

* Every compact space is countably compact.
*A countably compact space is compact if and only if it is Lindelöf.
*Every countably compact space is limit point compact.
*For T1 spaces, countable compactness and limit point compactness are equivalent.
*Every sequentially compact space is countably compact. The converse does not hold. For example, the product of continuum-many closed intervals ,1/math> with the product topology is compact and hence countably compact; but it is not sequentially compact.
*For first-countable spaces, countable compactness and sequential compactness are equivalent.
*For metrizable spaces, countable compactness, sequential compactness, limit point compactness and compactness are all equivalent.
*The example of the set of all real numbers with the standard topology shows that neither local compactness nor σ-compactness nor paracompactness imply countable compactness.
*Closed subspaces of a countably compact space are countably compact.
*The continuous image of a countably compact space is countably compact.
*Every countably compact space is pseudocompact.
*In a countably compact space, every locally finite family of nonempty subsets is finite.
*Every countably compact paracompact space is compact.
* Every countably compact Hausdorff first-countable space is regular.
*Every normal countably compact space is collectionwise normal.
*The product of a compact space and a countably compact space is countably compact.
*The product of two countably compact spaces need not be countably compact.Engelking, example 3.10.19, p. 205

See also

* Sequentially compact space
* Compact space
* Limit point compact
*Lindelöf space

Notes

References

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* {{Citation , last=Willard , first=Stephen , title=General Topology , orig-year=1970 , publisher=Addison-Wesley , edition=Dover reprint of 1970 , year=2004

Properties of topological spaces