All cluster points of countable sets in supercompact spaces are the limits of nontrivial sequences
Author:
Zhong Qiang Yang
Journal:
Proc. Amer. Math. Soc. 122 (1994), 591595
MSC:
Primary 54D30
MathSciNet review:
1209102
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Abstract: A space is called supercompact if it has an open subbase such that every cover consisting of elements of the subbase has a subcover consisting of two elements. In this paper we prove that, in a continuous image of a closed set of a supercompact space, a point is a cluster point of a countable set if and only if it is the limit of a nontrivial sequence. As corollaries, we answer questions asked by J. van Mill et al.
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 M. G. Bell, Supercompactness of compactification and hyperspaces, Trans. Amer. Math. Soc. 281 (1984), 717724. MR 722770 (84m:54022)
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 , Not all dyadic spaces are supercompact, Comment. Math. Univ. Carolin. 31 (1990), 775779. MR 1091375 (92c:54029)
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 , A first countable supercompact Hausdoff space with a closed nonsupercompact subspace, Colloq. Math. 43 (1980), 233241. MR 628178 (83h:54033)
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 W. Bula, J. Nikiel, H. M. Tuncali, and E. D. Tymchatyn, Continuous images of ordered compacta are regular supercompact, Topology Appl. 45 (1992), 203221. MR 1180810 (93i:54015)
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 E. K. van Douwen, Special bases for compact metrizable spaces, Fund. Math. 61 (1981), 201209. MR 611760 (82d:54036)
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 E. K. van Douwen and J. van Mill, Supercompact spaces, Topology Appl. 13 (1982), 2132. MR 637424 (82m:54017)
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 J. de Groot, Supercompactness and superextension, Contribution to Ext. Theory Top. Struct. Symposium (Berlin), Dentyscher Verlaae der Wissenchafen, Berlin, 1969, pp. 8990.
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 J. van Mill, Supercompactness and Wallman spaces, Math. Centre Tract, vol. 85, NorthHolland, Amsterdam, 1977. MR 0464160 (57:4095)
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 , A nonsupercompact continuous image of a supercompact space, Houston J. Math. 5 (1979), 241247. MR 546758 (80m:54033)
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 C. F. Mills, A simpler proof that compact metric spaces are supercompact, Proc. Amer. Math. Soc. 73 (1979), 388390. MR 518526 (80d:54036)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199412091020
PII:
S 00029939(1994)12091020
Keywords:
Supercompact,
limit,
sequence
Article copyright:
© Copyright 1994
American Mathematical Society
