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][2]
- (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.
Examples
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.[4] The converse does not hold. For example, the product of continuum-many closed intervals with the product topology is compact and hence countably compact; but it is not sequentially compact.[5]
- For first-countable spaces, countable compactness and sequential compactness are equivalent.[6] More generally, the same holds for sequential spaces.[7]
- 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.[8]
- The continuous image of a countably compact space is countably compact.[9]
- Every countably compact space is pseudocompact.
- In a countably compact space, every locally finite family of nonempty subsets is finite.[11]
- Every countably compact paracompact space is compact.[11] More generally, every countably compact metacompact space is compact.
- Every countably compact Hausdorff first-countable space is regular.[14][15]
- Every normal countably compact space is collectionwise normal.
- The product of a compact space and a countably compact space is countably compact.[16][17]
- The product of two countably compact spaces need not be countably compact.[18]
See also
Notes
- ^ Steen & Seebach, p. 19
- ^ "General topology - Does sequential compactness imply countable compactness?".
- ^ Steen & Seebach, p. 20
- ^ Steen & Seebach, Example 105, p, 125
- ^ Willard, problem 17G, p. 125
- ^ Kremsater, Terry Philip (1972), Sequential space methods (Thesis), University of British Columbia, doi:10.14288/1.0080490, Theorem 1.20
- ^ Willard, problem 17F, p. 125
- ^ Willard, problem 17F, p. 125
- ^ a b "Countably compact paracompact space is compact".
- ^ Steen & Seebach, Figure 7, p. 25
- ^ "Prove that a countably compact, first countable T2 space is regular".
- ^ Willard, problem 17F, p. 125
- ^ "Is the Product of a Compact Space and a Countably Compact Space Countably Compact?".
- ^ Engelking, example 3.10.19
References