Given a partially ordered set $P$ we study properties of topological spaces $X$ admitting a $P$-base, i.e., an indexed family $(U_��)_{��\in P}$ of subsets of $X\times X$ such that $U_��\subset U_��$ for all $��\le��$ in $P$ and for every $x\in X$ the family $(U_��[x])_{��\in P}$ of balls $U_��[x]=\{y\in X:(x,y)\in U_��\}$ is a neighborhood base at $x$. A $P$-base $(U_��)_{��\in P}$ for $X$ is called locally uniform if the family of entourages $(U_��U_��^{-1}U_��)_{��\in P}$ remains a $P$-base for $X$. A topological space is first-countable if and only if it has an $��$-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 $��$-base. In the paper we shall study topological spaces possessing a (locally uniform) $��^��$-base. Our results show that spaces with an $��^��$-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 $��^��$-based topological spaces. On the other hand, topological spaces with a locally uniform $��^��$-base have many properties, typical for generalized metric spaces. Also we study Tychonoff spaces whose universal (pre- or quasi-) uniformity has an $��^��$-base and show that such spaces are close to being $��$-compact.

105 pages