Cardinalities of first countable -closed spaces

Authors:
Alan Dow and Jack Porter

Journal:
Proc. Amer. Math. Soc. **89** (1983), 527-532

MSC:
Primary 54D25; Secondary 54A25

DOI:
https://doi.org/10.1090/S0002-9939-1983-0715880-X

MathSciNet review:
715880

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Abstract: It is now well known that first countable compact Hausdorff spaces are either countable or have cardinality . The situation for first countable -closed spaces is that they have cardinality less than or equal , and it is at least consistent that they may have cardinality . We show that the situation is quite different for first countable -closed spaces. We begin by constructing an example which has cardinality . Let be the smallest cardinal greater than which is not a successor. For each cardinal with we construct a first countable -closed space of cardinality . We also construct a first countable -closed space of cardinality . This seems to indicate that there is no reasonable upper bound to the cardinalities of -closed spaces as a function of their character.

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DOI:
https://doi.org/10.1090/S0002-9939-1983-0715880-X

Article copyright:
© Copyright 1983
American Mathematical Society