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) [math]\displaystyle{ \Rightarrow }[/math] (2): Suppose (1) holds and A is an infinite subset of X without [math]\displaystyle{ \omega }[/math]-accumulation point. By taking a subset of A if necessary, we can assume that A is countable. Every [math]\displaystyle{ x\in X }[/math] has an open neighbourhood [math]\displaystyle{ O_x }[/math] such that [math]\displaystyle{ O_x\cap A }[/math] is finite (possibly empty), since x is not an ω-accumulation point. For every finite subset F of A define [math]\displaystyle{ O_F = \cup\{O_x: O_x\cap A=F\} }[/math]. Every [math]\displaystyle{ O_x }[/math] is a subset of one of the [math]\displaystyle{ O_F }[/math], so the [math]\displaystyle{ O_F }[/math] cover X. Since there are countably many of them, the [math]\displaystyle{ O_F }[/math] form a countable open cover of X. But every [math]\displaystyle{ O_F }[/math] intersect A in a finite subset (namely F), so finitely many of them cannot cover A, let alone X. This contradiction proves (2). (2) [math]\displaystyle{ \Rightarrow }[/math] (3): Suppose (2) holds, and let [math]\displaystyle{ (x_n)_n }[/math] 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 [math]\displaystyle{ A=\{x_n: n\in\mathbb N\} }[/math] 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) [math]\displaystyle{ \Rightarrow }[/math] (1): Suppose (3) holds and [math]\displaystyle{ \{O_n: n\in\mathbb N\} }[/math] is a countable open cover without a finite subcover. Then for each [math]\displaystyle{ n }[/math] we can choose a point [math]\displaystyle{ x_n\in X }[/math] that is not in [math]\displaystyle{ \cup_{i=1}^n O_i }[/math]. The sequence [math]\displaystyle{ (x_n)_n }[/math] has an accumulation point x and that x is in some [math]\displaystyle{ O_k }[/math]. But then [math]\displaystyle{ O_k }[/math] is a neighborhood of x that does not contain any of the [math]\displaystyle{ x_n }[/math] with [math]\displaystyle{ n\gt k }[/math], so x is not an accumulation point of the sequence after all. This contradiction proves (1). (4) [math]\displaystyle{ \Leftrightarrow }[/math] (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 [math]\displaystyle{ [0,1] }[/math] 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.
• 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.
• 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

References

• Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Berlin. ISBN 3-88538-006-4. 
• James Munkres (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2. 
• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover reprint of 1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. 
• Willard, Stephen (2004), General Topology (Dover reprint of 1970 ed.), Addison-Wesley