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

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 $U_\alpha[x]={y\in X:(x,y)\in U_\alpha}$isaneighborhoodbaseat$x$\mathrm{.}A$P$-base$(U_\alpha){\alpha\in P}$for$X$iscalledlocallyuniformifthefamilyofentourages$(U\alpha U_\alpha^{-1}U_\alpha)_{\alpha\in P}$remainsa$P$-basefor$X$. A topological space is first-countable if and only if it has an $\omega$-base. By Moore’s Metrization Theorem, a topological space is metrizable if and only if it is a $T_0$-space with a locally uniform $\omega$-base. In the paper we shall 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.