Products of countably compact spaces

Abstract

Abstract.Extensions of sufficient conditions for the product of two countably compact spaces to be countably compact, plus a relevant example.In 1953 Novak [4] published an example to show that countable compactness is not preserved under products. Novak's example consists of taking two countably compact subspaces Ax and A2 of the Stone-Cech compactification ßN of the natural numbers A such that Ax U A2 = ßN and Ax n A2 = A; the product Ax X A2 is not countably compact because it contains an infinite closed discrete space.Additional conditions are thus necessary on one of the countably compact spaces A or F to ensure countable compactness of the product A x F. Some of the additional properties on A which will guarantee this are: sequentially compact, first countable, sequential, k. These properties and some proofs have been discussed in a paper of S. Franklin [2]. Other properties which generate countably compact products in this manner are paracompactness and metacompactness, since either of these conditions, when added to countable compactness of a factor, makes the factor compact, and the product of a compact space with a countably compact space is well known to be countably compact. In what follows, assume all spaces Hausdorff.A space which is a generalization of /:-space (hence, of first countable and sequential space) has proved fruitful in some product theorems. This space is called a weakly-^ space. Definition. A topological space A is weakly-/: iff a subset F of A is closed in X if F fi C is finite for every compact C in A.This notation was introduced in [5]. Weakly-A: spaces can be used to generalize the results mentioned above.Theorem. If X and Y are countably compact spaces such that X is weakly-k, then the product X X Y is countably compact.Proof. Consider the sequence {(xn,yn): n G A} in A X F. If the sequence {xn} is closed, we are done, because there is a subsequence of [xn] with a limit point in A. So assume xn # xm for all m ¥= « and that [xn] is not closed. By the weakly-Ä: condition a compact C exists in A such that card (C n {-*"})

The topological properties of q-spaces in free topological groups

“…Then X is countably compact since it is collectionwise Hausdorff. Therefore, it follows from [12] that X 2 is countably compact.…”

“…(b) Each weakly-k cf -space is discrete. However, there exists a weakly-k space which is not a k-space, see [12]. Obviously, each collectionwise Hausdorff pseudocompact space is countably compact.…”