The Number of Closed Subsets of a Topological Space

Abstract

Let X be an infinite topological space, let 𝔫 be an infinite cardinal number with 𝔫 ≦ |X|. The basic problem in this paper is to find the number of closed sets in X of cardinality 𝔫. A complete answer to this question for the class of metrizable spaces has been given by A. H. Stone in [31], where he proves the following result. Let X be an infinite metrizable space of weight 𝔪, let 𝔫 ≦ |X|.

Countably compact, locally countable 𝑇₂-spaces

“…This example answers a question raised several years ago in a conversation with R. E. Hodel. Hodel has just proved [9] that if P is an infinite space in which every point is a Hausdorff-Gs then there are \P\" countable, closed subsets of P. The question was raised whether the conclusion of this result still holds in 'T2-spaces in which every point is a G$". Example (D) shows that this is not the case.…”