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Spaces with \(\sigma\)-locally finite Lindelöf sn-networks. (English) Zbl 1313.54058

Summary: We prove that a space \(X\) has a \(\sigma\)-locally finite Lindelöf sn-network if and only if \(X\) is a compact-covering compact and mssc-image of a locally separable metric space, if and only if \(X\) is a sequentially-quotient \(\pi\) and mssc-image of a locally separable metric space, where “compact-covering” (or “sequentially-quotient”) can not be replaced by “sequence-covering”. As an application, we give a new characterization of spaces with locally countable weak bases.

MSC:

54E35 Metric spaces, metrizability
54E40 Special maps on metric spaces
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