In
mathematics a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
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)
(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.
Examples
*The
first uncountable ordinal (with the
order topology
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.
If ''X'' is a totally ordered set, th ...
) is an example of a countably compact space that is not compact.
Properties
* Every
compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", ...
is countably compact.
*A countably compact space is compact if and only if it is
Lindelöf.
*Every countably compact space is
limit point compact In mathematics, a topological space ''X'' is said to be limit point compact or weakly countably compact if every infinite subset of ''X'' has a limit point in ''X''. This property generalizes a property of compact spaces. In a metric space, limit ...
.
*For
T1 space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. An R0 space is one in which this holds for every pair of to ...
s, countable compactness and limit point compactness are equivalent.
*Every
sequentially compact space
In mathematics, a topological space ''X'' is sequentially compact if every sequence of points in ''X'' has a convergent subsequence converging to a point in X.
Every metric space is naturally a topological space, and for metric spaces, the noti ...
is countably compact. The converse does not hold. For example, the product of
continuum-many closed intervals