$ω^ω$-bases in topological and uniform spaces

Taras Banakh

Published 2016-07-27Version 1

We study topological properties of topological spaces with a local $\omega^\omega$-base and uniform spaces with an $\omega^\omega$-base. A topological space $X$ is defined to have a local $\omega^\omega$-base if each point of $X$ has a neighborhood base $\{U_\alpha\}_{\alpha\in\omega^\omega}$ such that $U_\beta\subset U_\alpha$ for all $\alpha\le\beta$ in $\omega^\omega$. A uniform space $X$ is defined to have an $\omega^\omega$-base if it has a base $\{U_\alpha\}_{\alpha\in\omega^\omega}$ of the uniformity such that $U_\beta\subset U_\alpha$ for all $\alpha\le\beta$ in $\omega^\omega$. Our results show that topological spaces with a local $\omega^\omega$-base and uniform spaces with an $\omega^\omega$-base possess many properties, typical for generalized metric spaces.