$\pi$-metrizable spaces and strongly $\pi$-metrizable spaces

zbMATH Open1273.54024arXiv1302.4192MaRDI QIDQ4923738

Shou Lin, Fucai Lin

Publication date: 27 May 2013

Abstract: A space $X$ is said to be $pi$-metrizable if it has a $sigma$-discrete $pi$-base. In this paper, we mainly give affirmative answers for two questions about $pi$-metrizable spaces. The main results are that: (1) A space $X$ is $pi$-metrizable if and only if $X$ has a $sigma$-hereditarily closure-preserving $pi$-base; (2) $X$ is $pi$-metrizable if and only if $X$ is almost $sigma$-paracompact and locally $pi$-metrizable; (3) Open and closed maps preserve $pi$-metrizability; (4) $pi$-metrizability satisfies hereditarily closure-preserving regular closed sum theorems. Moreover, we define the notions of second-countable $pi$-metrizable and strongly $pi$-metrizable spaces, and study some related questions. Some questions about strongly $pi$-metrizability are posed.

Full work available at URL: https://arxiv.org/abs/1302.4192

zbMATH Keywords

perfect map\(\pi\)-baseclosed mapFréchet spaceseparable spaceirreducible map\(\pi\)-weightquasi-open map\(\pi\)-metrizable spacealmost \(\sigma\)-paracompact spacelocally \(\pi\)-metrizable spacestrong \(\pi\)-basestrongly \(\pi\)-metrizable spacestrongly \(d\)-separable space

Mathematics Subject Classification ID

Special maps on topological spaces (open, closed, perfect, etc.) (54C10) Noncompact covering properties (paracompact, Lindelöf, etc.) (54D20) Base properties of topological spaces (54D70) Separability of topological spaces (54D65)

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