On 𝑘-spaces

Abstract

One of our first interests in the study of k-spaces (spaces with the weak (= finest) topology generated by their compact subsets) was in the determination of those subsets of a topological space which are themselves k-spaces. If a subset A of a space X is a k-space, thein a subset of A is closed in A if it intersects every compact subset K of X in a set closed in AQnK. If a subset A which is not necessarily a k-space has this property relative to the compact subsets of X, then we say that A has property (k). Throughout this paper, unless exception is noted, we shall assume that compact subsets are closed. A subspace A of a space X is a k-space if and only if A has property (k) and A nK is a k-space for each compact subset K of X (Theorem 1). It follows that an open subset of a k-space in which compact subsets are is itself a k-space. The case when the k-space is Hausdorff was first established independently in [6] by a different proof. This result is similar to a theorem of N. E. Steenrod [7] (an open subset of a k-space is a k-space if it is a regular open set). The class of subspaces of a space X having property (k) is of interest in itself. For example, every subset of X has property (k) if and only if for any set A CX and any point xEA, there exists a compact set K such that xCAnK (Theorem 2). A. Arhangel'skil [1] calls spaces with this property k'-spaces. Analogous to this theorem is a result proved by Arhangel'skil [2] which states that a space X is an hereditary k-space if and only if X is a Frechet-Urysohn space.