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It is shown that certain weak-base structures on a topological space give a -space. This solves the question by A.V. Arhangel’skii of when quotient images of metric spaces are -spaces. A related result about symmetrizable spaces also answers a question of Arhangel’skii.
Hence, quotient mappings, with compact fibers, from metric spaces have a -space image. What about quotient -mappings? Arhangel’skii and Buzyakova have shown that spaces with a point-countable base are -spaces...
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