Realcompactness of hyperspaces and extensions of open-closed maps |
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Authors: | Haruto Ohta |
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Affiliation: | Faculty of Education, Shizuoka University, Ohya, Shizuoka, 422, Japan |
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Abstract: | In response to questions of Ginsburg [9, 10], we prove that if cf(c)>ω1, then there exists an open-closed, continuous map f from a normal, realcompact space X onto a space Y which is not realcompact. By his result the hyperspace 2x of closed subsets of X is then not realcompact, and the extension μf(vf) of f to the topological completion (the Hewitt realcompactification) of X is not onto. The latter fact solves problems raised by Morita [16] and by Isiwata [12] both negatively. We also consider the problem whether or not the hyperspace of a hereditarily Lindelöf space is hereditarily realcompact. |
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Keywords: | 54B20, 54D60, 54C10 54D20 |
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