All spaces under consideration are Hausdorff. Countably compactness and sequentially compactness are notions related to compactness. A countably compact space is one in which every countable open cover has a finite subcover, or equivalently, every countably infinite subset has a limit point. For a space , the point is a limit point of if every open subset of containing contains a point of distinct from . On the other hand, a space is sequentially compact if every sequence of points of has a subsequence that converges. Any sequentially compact space is countably compact. The converse is not true. The product space where is not sequentially compact (see Sequentially compact spaces, I) . However, for sequential spaces (first countable spaces in particular), the notion of sequentially compactness and countably compactness are equivalent. For previous discussion in this blog about sequential spaces, see the links below.
Any countably compact space that is countable in size is metrizable and thus first countable.
Proof. Let be countably compact such that . Then is compact (any Lindelof countably compact space is compact). In any countable space, the set of all singleton sets is a countable network. Any compact Hausdorff space with a countable network is metrizable and thus first countable. See Spaces With Countable Network.
Let be a sequential space. Then is countably compact if and only if is sequentially compact.
Proof. The direction always holds without the space being sequential.
Suppose is countably compact. Suppose that is not sequentially compact. Then there is a sequence of points of with no convergent subsequence. Let be the set of all terms in this sequence, i.e. . Note that is sequentially closed. Since is sequential, is closed in . As a closed subset of a countably compact space, is countably compact. By the lemma, is first countable. Since is an infinite compact space, has a non-isolated point . This means some sequence of points of converges to , contradicting the assumption that has no convergent subsequence. Therefore must be sequentially compact.
Previous posts on sequential spaces and k-spaces:
Sequential spaces, I
Sequential spaces, II
Sequential spaces, III
Sequential spaces, IV
Sequential spaces, V
A note about the Arens’ space
An observation about sequential spaces
- Engelking, R. General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
- Henkel, D. Solution to Monthly Problem 5698, American Mathematical Monthly 77, p. 896, 1970
- Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.