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Joint metrizability of subspaces and perfect mappings. (English) Zbl 1294.54005

The joint metrizability in the title refers to the possibility of having a single metric on a (non-metrizable) topological space that induces the topology on all members of a specified class of metrizable subspaces, e.g., the discrete metric will work for all relatively discrete subspaces. The authors define a space to be JPM if one can get one metric to work for all metrizable subspaces. Some sample results are: a \(k\)-space is metrizable iff there is a metric that metrizes all compact and all countable closed discrete subspaces; a regular and first-countable JPM-space is metrizable (hence the Sorgenfrey line is not JPM). As the standard map from the absolute to a Tychonoff space shows, the JPM property is not preserved by perfect maps as extremally disconnected spaces are trivially JPM. If the fibers are sequential then the image is JPM.
Reviewer: K. P. Hart (Delft)

MSC:

54B05 Subspaces in general topology
54A05 Topological spaces and generalizations (closure spaces, etc.)
54E35 Metric spaces, metrizability
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