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The Number of Closed Subsets of a Topological Space

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|.