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• W2524289452 abstract "Given a partially ordered set $P$ we study properties of topological spaces $X$ admitting a $P$-base, i.e., an indexed family $(U_alpha)_{alphain P}$ of subsets of $Xtimes X$ such that $U_betasubset U_alpha$ for all $alphalebeta$ in $P$ and for every $xin X$ the family $(U_alpha[x])_{alphain P}$ of balls $U_alpha[x]={yin X:(x,y)in U_alpha}$ is a neighborhood base at $x$. A $P$-base $(U_alpha)_{alphain P}$ for $X$ is called locally uniform if the family of entourages $(U_alpha U_alpha^{-1}U_alpha)_{alphain P}$ 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 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." @default.
• W2524289452 created "2016-10-07" @default.
• W2524289452 creator A5088614199 @default.
• W2524289452 date "2016-07-27" @default.
• W2524289452 modified "2023-09-27" @default.
• W2524289452 title "Topological spaces with an $omega^omega$-base" @default.
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