Cardinalities of first countable closed spaces
Authors:
Alan Dow and Jack Porter
Journal:
Proc. Amer. Math. Soc. 89 (1983), 527532
MSC:
Primary 54D25; Secondary 54A25
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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 [DP1]
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 , Cardinalities of closed spaces, Topology Proc. 7 (1982).
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 A. A. Gryzlow, Two theorems on the cardinality of topological spaces, Soviet Math. Dokl. 21 (1980), 506509.
 [H]
 S. H. Hechler, Two closed spaces revisited, Proc. Amer. Math. Soc. 56 (1976), 303309. MR 0405354 (53:9148)
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 H. Herrlich, Abgeschlossenheit und Minimalität, Math. Z. 88 (1965), 285294. MR 0184191 (32:1664)
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 D. Pettey, Products of regularclosed spaces, Topology Appl. 14 (1982), 189199. MR 667666 (84h:54022)
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 R. M. Stephenson, Jr., Two closed spaces, Canad. J. Math. 24 (1972), 286292. MR 0298613 (45:7665)
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 J. E. Vaughan, A countably compact, first countable, nonnormal space, Proc. Amer. Math. Soc. 15 (1979), 339342. MR 532163 (80e:54022)
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DOI:
http://dx.doi.org/10.1090/S0002993919830715880X
PII:
S 00029939(1983)0715880X
Article copyright:
© Copyright 1983
American Mathematical Society
