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Concept# Countably compact space

Summary

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

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