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.

Proof of equivalence

(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 $A=\{x_n:n\in \mathbb{\mathbb{N}}\}$ 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 $\{O_n:n\in \mathbb{\mathbb{N}}\}$ 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 $\underset{i=1}{\overset{n}{\cup }}{O}_i$. 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) $\iff$ (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.

See also

• Sequentially compact space
• Compact space
• Limit point compact
• Lindelöf space