$\mathfrak P_0$-spaces

TL;DR

This paper introduces $rak P_0$-spaces, a class of regular topological spaces with countable Pytkeev networks, exploring their properties, examples, and stability under various topological operations.

Contribution

It defines $rak P_0$-spaces, studies their properties, and shows their stability under operations like subspaces, products, and function spaces, extending the class of known topological spaces with these features.

Findings

$rak P_0$-spaces include all metrizable separable spaces.

Closed under subspaces, countable products, and certain limits.

Function spaces and free topological groups over $rak P_0$-spaces are also $rak P_0$.

Abstract

A regular topological space $X$ is defined to be a $\mathfrak P_0$-space if it has countable Pytkeev network. A network $\mathcal N$ for $X$ is called a Pytkeev network if for any point $x\in X$, neighborhood $O_x\subset X$ of $x$ and subset $A\subset X$ accumulating at a $x$ there is a set $N\in\mathcal N$ such that $N\subset O_x$ and $N\cap A$ is infinite. The class of $\mathfrak P_0$-spaces contains all metrizable separable spaces and is (properly) contained in the Michael's class of $\aleph_0$-spaces. It is closed under many topological operations: taking subspaces, countable Tychonoff products, small countable box-products, countable direct limits, hyperspaces of compact subsets. For an $\aleph_0$-space $X$ and a $\mathfrak P_0$-space $Y$ the function space $C_k(X,Y)$ endowed with the compact-open topology is a $\mathfrak P_0$-space. For any sequential $\aleph_0$-space $X$ the free abelian topological group $A(X)$ and the free locally convex linear topological space $L(X)$ both are $\mathfrak P_0$-spaces. A sequential space is a $\mathfrak P_0$-space if and only if it is an $\aleph_0$-space. A topological space is metrizable and separable if and only if it is a $\mathfrak P_0$-space with countable fan tightness.