Countably compact space

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 $$O_F = \cup\{O_x: O_x\cap A=F\}$$. 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 $$A=\{x_n: n\in\mathbb N\}$$ 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 $$\{O_n: n\in\mathbb N\}$$ 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 $$\cup_{i=1}^n O_i$$. 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 $$[0,1]$$ 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. More generally, the same holds for sequential spaces.
 * 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. More generally, every countably compact metacompact 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.