Sequentially compact spaces, II

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 X, the point p \in X is a limit point of A \subset X if every open subset of X containing p contains a point of A distinct from p. On the other hand, a space X is sequentially compact if every sequence \left\{x_n:n=1,2,3,\cdots\right\} of points of X has a subsequence that converges. Any sequentially compact space is countably compact. The converse is not true. The product space 2^I where I=[0,1] 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.

Lemma
Any countably compact space that is countable in size is metrizable and thus first countable.

Proof. Let X be countably compact such that \lvert X \lvert=\aleph_0. Then X 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. \blacksquare

Theorem
Let X be a sequential space. Then X is countably compact if and only if X is sequentially compact.

Proof. The direction \Leftarrow always holds without the space being sequential.

\Rightarrow Suppose X is countably compact. Suppose that X is not sequentially compact. Then there is a sequence \left\{x_n\right\} of points of X with no convergent subsequence. Let A be the set of all terms in this sequence, i.e. A=\left\{x_n:n=1,2,3,\cdots\right\}. Note that A is sequentially closed. Since X is sequential, A is closed in X. As a closed subset of a countably compact space, A is countably compact. By the lemma, A is first countable. Since A is an infinite compact space, A has a non-isolated point x. This means some sequence of points of A converges to x, contradicting the assumption that \left\{x_n\right\} has no convergent subsequence. Therefore X must be sequentially compact. \blacksquare

Previous posts on sequential spaces and k-spaces:
Sequential spaces, I
Sequential spaces, II
Sequential spaces, III
Sequential spaces, IV
Sequential spaces, V
k-spaces, I
k-spaces, II
A note about the Arens’ space
An observation about sequential spaces

Reference

  1. Engelking, R. General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
  2. Henkel, D. Solution to Monthly Problem 5698, American Mathematical Monthly 77, p. 896, 1970
  3. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.
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