- (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) (2): Suppose (1) holds and A is an infinite subset of X without -accumulation point. By taking a subset of A if necessary, we can assume that A is countable. Every has an open neighbourhood such that is finite (possibly empty), since x is not an ω-accumulation point. For every finite subset F of A define . Every is a subset of one of the , so the cover X. Since there are countably many of them, the form a countable open cover of X. But every intersect A in a finite subset (namely F), so finitely many of them cannot cover A, let alone X. This contradiction proves (2).
(2) (3): Suppose (2) holds, and let 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 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) (1): Suppose (3) holds and is a countable open cover without a finite subcover. Then for each we can choose a point that is not in . The sequence has an accumulation point x and that x is in some . But then is a neighborhood of x that does not contain any of the with , so x is not an accumulation point of the sequence after all. This contradiction proves (1).
(4) (1): Conditions (1) and (4) are easily seen to be equivalent by taking complements.
- The first uncountable ordinal (with the order topology) is an example of a countably compact space that is not compact.
- Every compact space is countably compact.
- A countably compact space is compact if and only if it is Lindelöf.
- A countably compact space is always limit point compact.
- For T1 spaces, countable compactness and limit point 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.
- 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 normal countably compact space is collectionwise normal.