The strong Pytkeev property in topological spaces

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 $(X_n,*_n)$, $n\inω$, 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 $k_ω$-subspace. A locally precompact topological group is metrizable if and only if it contains a dense subgroup with the strong Pytkeev property.