The strong Pytkeev property in topological spaces

TL;DR

This paper investigates the strong Pytkeev property in various topological spaces, establishing conditions under which function spaces, products, and groups possess this property, and characterizing sequential rectifiable spaces with it.

Contribution

It proves that function spaces over -spaces and spaces with the strong Pytkeev property inherit this property, and characterizes sequential rectifiable spaces with the strong Pytkeev property.

Findings

Function spaces $C_k(X,Y)$ have the strong Pytkeev property under certain conditions.

Tychonoff and small box-products of spaces with the strong Pytkeev property also have it.

Metrizability of groups relates to the presence of dense subgroups with the strong Pytkeev property.

Abstract

A topological space $X$ has the strong Pytkeev property at a point $x\in X$ if there exists a countable family $\mathcal N$ of subsets of $X$ such that for each neighborhood $O_x\subset X$ and subset $A\subset X$ accumulating at $x$, there is a set $N\in\mathcal N$ such that $N\subset O_x$ and $N\cap A$ is infinite. We prove that for any $\aleph_0$-space $X$ and any space $Y$ with the strong Pytkeev property at a point $y\in Y$ the function space $C_k(X,Y)$ has the strong Pytkeev property at the constant function $X\to \{y\}\subset Y$. If the space $Y$ is rectifiable, then the function space $C_k(X,Y)$ is rectifiable and has the strong Pytkeev property at each point. We also prove that for any pointed spaces $(X_n,*_n)$, $n\in\omega$, with the strong Pytkeev property their Tychonoff product and their small box-product both have the strong Pytkeev property at the distinguished point. We prove that a sequential rectifiable space $X$ has the strong Pytkeev property if and only if $X$ is metrizable or contains a clopen submetrizable $k_\omega$-subspace. A locally precompact topological group is metrizable if and only if it contains a dense subgroup with the strong Pytkeev property.