Topological spaces with a local $ω^ω$-base have the strong Pytkeev$^*$ property

Taras Banakh

Published 2016-07-13Version 1

Modifying the known definition of a Pytkeev network, we introduce a notion of Pytkeev$^*$ network and prove that a topological space has a countable Pytkeev network at a point $x\in X$ if and only if $X$ is countably tight at $x$ and has a countable Pykeev$^*$ network at $x$. We define a topological space $X$ to have the strong Pytkeev$^*$ property if $X$ has a Pytkeev$^*$ network at each point. Our main theorem says that a topological space $X$ has a countable Pytkeev$^*$ network at point $x\in X$ if $X$ has a local $\omega^\omega$-base at $x$ (i.e., a neighborhood base $(U_\alpha)_{\alpha\in\omega^\omega}$ at $x$ such that $U_\beta\subset U_\alpha$ for all $\alpha\le\beta$ in $\omega^\omega$). Consequently, each countably tight space with a local $\omega^\omega$-base has the strong Pytkeev property.