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:

 5.4e+36 Metric spaces, metrizability 5.4e+41 Special maps on metric spaces
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