On PP-spaces and related concepts

The concept of the strong Pytkeev property, recently introduced by Tsaban and Zdomskyy in [32], was successfully applied to the study of the space Cc(X)Cc(X) of all continuous real-valued functions with the compact-open topology on some classes of topological spaces X   including Čech-complete Lindelöf spaces. Being motivated also by several results providing various concepts of networks we introduce the class of PP-spaces strictly included in the class of ℵ-spaces. This class of generalized metric spaces is closed under taking subspaces, topological sums and countable products and any space from this class has countable tightness. Every PP-space X has the strong Pytkeev property. The main result of the present paper states that if X   is an ℵ0ℵ0-space and Y   is a PP-space, then the function space Cc(X,Y)Cc(X,Y) has the strong Pytkeev property. This implies that for a separable metrizable space X and a metrizable topological group G   the space Cc(X,G)Cc(X,G) is metrizable if and only if it is Fréchet–Urysohn. We show that a locally precompact group G   is a PP-space if and only if G is metrizable.