Category in function spaces. I

Abstract

In this paper we study Baire category in spaces of continuous, real-valued functions equipped with the topology of pointwise convergence. We show that, for normal spaces, the Baire category of Cπ(X) is determined by the Baire category of Cπ(Y) for certain small subspaces Y of X and that the category of Cπ(X) is intimately related to the existence of winning strategies in a certain topological game Γ(X) played in the space X. We give examples of certain countable regular spaces for which Cπ(X) is a Baire space and we characterize those spaces X for which Cπ(X) has one of the stronger completeness properties, such as pseudocomplet eness or Cech-completeness. l Introduction* It is well-known that the set of all continuous real-valued functions on a space X is a completely metrizable space when equipped with the topology of uniform convergence and therefore that the Baire Category Theorem is valid in this space. However, when function spaces carry other topologies, the status and role of the Baire category property is more mysterious, and in this paper we consider Baire category in Cπ{X), the set of all continuous real-valued functions on a completely Hausdorff space X, equipped with the topology of pointwise convergence. (See § 2 for precise definitions.) We can preview some of our less technical results as follows. Experience shows that the Baire category of Cπ(X) is determined by the kinds of limit points which countable subsets of X can have. For example, if Cπ(X) is a Baire space then X has no nontrivial convergent sequences (Corollary 3.3) and the same argument shows that the set of bounded members of Cπ(X) is never a Baire space. Furthermore, it is sometimes possible to study the Baire category of Cπ(X) by examining the continuously extendable functions defined on countable subspaces of X (3.7). Examples show that there are infinite nondiscrete spaces X for which Cπ(X) is a Baire space, e.g., any normal space in which each countable subset is closed (see Theorem 8.4). A more interesting fact is that there are even countable nondiscrete spaces for which Cπ(X) is a Baire space, e.g., any space X = ω U {p} where p e βω — ω, topologized as a subspace of βω (see Example 7.1). It is possible for Cπ(X) to be a Baire space, where X is countable and regular, without X being embeddable in βω (Example 7.2) and we can characterize those filters ^V on ω such <that Gπ{X) is a Baire space when X is obtained by adjoining a single point to ω and using Λr as the neighborhood filter of the ideal point. More precisely, if X has a unique nonisolated