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 $(X_n,{\ast }_n)$, $n\in \omega$, 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.

15 pages. arXiv admin note: text overlap with arXiv:1311.1468

References in corpus (1)

• Sequential rectifiable spaces of countable $c{s}^{\ast }$-character

Cited by in corpus (3)

• On $\mathfrak{P}$-spaces and related concepts
• Countable tightness and $\mathfrak{G}$-bases on Free topological groups
• On topological spaces and topological groups with certain local countable networks