On $\mathfrak{P}$-spaces and related concepts

TL;DR

This paper introduces the class of -spaces, explores their properties, and demonstrates that certain function spaces involving these spaces have the strong Pytkeev property, with implications for metrizable topological groups.

Contribution

It defines -spaces, shows they are closed under key operations, and proves that function spaces with -spaces have the strong Pytkeev property, extending understanding of these spaces.

Findings

-spaces are closed under subspaces, sums, and products.

Function spaces C_c(X,Y) have the strong Pytkeev property when X is -space and Y is -space.

Locally precompact -groups are exactly the metrizable ones.

Abstract

The concept of the strong Pytkeev property, recently introduced by Tsaban and Zdomskyy in [32], was successfully applied to the study of the space $C_c(X)$ of all continuous real-valued functions with the compact-open topology on some classes of topological spaces $X$ including \v{C}ech-complete Lindel\"{o}f spaces. Being motivated also by several results providing various concepts of networks we introduce the class of $\mathfrak{P}$-spaces strictly included in the class of $\aleph$-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 $\mathfrak{P}$-space $X$ has the strong Pytkeev property. The main result of the present paper states that if $X$ is an $\aleph_0$-space and $Y$ is a $\mathfrak{P}$-space, then the function space $C_c(X,Y)$ has the strong Pytkeev property. This implies that for a separable metrizable space $X$ and a metrizable topological group $G$ the space $C_c(X,G)$ is metrizable if and only if it is Fr\'{e}chet-Urysohn. We show that a locally precompact group $G$ is a $\mathfrak{P}$-space if and only if $G$ is metrizable.