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Topological spaces with an $\omega^{\omega}$-base

Volume 538 / 2019

Taras Banakh Dissertationes Mathematicae 538 (2019), 1-141 MSC: Primary 54D70, 54E15, 54E18, 54E35; Secondary 03E04, 03E17, 54A20, 54A25, 54A35, 54C35, 54D15, 54D45, 54D65, 54D70, 54G10, 54G20 DOI: 10.4064/dm762-4-2018 Published online: 18 March 2019


Given a partially ordered set $P$ we study properties of topological spaces $X$ admitting a $P$-base, i.e., an indexed family $(U_\alpha)_{\alpha\in P}$ of subsets of $X\times X$ such that $U_\beta\subset U_\alpha$ for all $\alpha\le\beta$ in $P$ and for every $x\in X$ the family $\{U_\alpha[x]\}_{\alpha\in P}$ of balls $U_\alpha[x]=\{y\in X:(x,y)\in U_\alpha\}$ is a neighborhood base at $x$. A $P$-base $(U_\alpha)_{\alpha\in P}$ for $X$ is called locally uniform if the family of entourages $(U_\alpha U_\alpha^{-1}U_\alpha)_{\alpha\in P}$ remains a $P$-base for $X$. A topological space is first-countable if and only if it has an $\omega$-base. By Moore’s Metrization Theorem, a $T_0$-space is metrizable if and only if it has a locally uniform $\omega$-base.

We study topological spaces possessing a (locally uniform) $\omega^\omega$-base. Our results show that spaces with an $\omega^\omega$-base share some common properties with first-countable spaces, in particular, many known upper bounds on the cardinality of first-countable spaces remain true for countably tight $\omega^\omega$-based topological spaces. On the other hand, topological spaces with a locally uniform $\omega^\omega$-base have many properties typical for generalized metric spaces. Also we study Tychonoff spaces whose universal (pre- or quasi-) uniformity has an $\omega^\omega$-base and show that such spaces are close to being $\sigma$-compact.


  • Taras BanakhInstitute of Mathematics
    Jan Kochanowski University in Kielce
    25-406 Kielce, Poland
    Faculty of Mechanics and Mathematics
    Ivan Franko National University in Lviv
    Lviv, 79000, Ukraine

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