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][2] (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.
has an open neighbourhood
is finite (possibly empty), since x is not an ω-accumulation point.
For every finite subset F of A define
form a countable open cover of X.
intersect A in a finite subset (namely F), so finitely many of them cannot cover A, let alone X.
This contradiction proves (2).
(3): Suppose (2) holds, and let
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.
(1): Suppose (3) holds and
is a countable open cover without a finite subcover.
we can choose a point
has an accumulation point x and that x is in some
, so x is not an accumulation point of the sequence after all.
This contradiction proves (1).
(1): Conditions (1) and (4) are easily seen to be equivalent by taking complements.