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), 591-595

MSC:
Primary 54D30

DOI:
https://doi.org/10.1090/S0002-9939-1994-1209102-0

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.

**[1]**M. G. Bell,*Supercompactness of compactification and hyperspaces*, Trans. Amer. Math. Soc.**281**(1984), 717-724. MR**722770 (84m:54022)****[2]**-,*Not all dyadic spaces are supercompact*, Comment. Math. Univ. Carolin.**31**(1990), 775-779. MR**1091375 (92c:54029)****[3]**-,*A first countable supercompact Hausdoff space with a closed**non-supercompact subspace*, Colloq. Math.**43**(1980), 233-241. MR**628178 (83h:54033)****[4]**W. Bula, J. Nikiel, H. M. Tuncali, and E. D. Tymchatyn,*Continuous images of ordered compacta are regular supercompact*, Topology Appl.**45**(1992), 203-221. MR**1180810 (93i:54015)****[5]**E. K. van Douwen,*Special bases for compact metrizable spaces*, Fund. Math.**61**(1981), 201-209. MR**611760 (82d:54036)****[6]**E. K. van Douwen and J. van Mill,*Supercompact spaces*, Topology Appl.**13**(1982), 21-32. MR**637424 (82m:54017)****[7]**J. de Groot,*Supercompactness and superextension*, Contribution to Ext. Theory Top. Struct. Symposium (Berlin), Dentyscher Verlaae der Wissenchafen, Berlin, 1969, pp. 89-90.**[8]**J. van Mill,*Supercompactness and Wallman spaces*, Math. Centre Tract, vol. 85, North-Holland, Amsterdam, 1977. MR**0464160 (57:4095)****[9]**J. van Mill and C. F. Mills,*Closed**subset of supercompact Hausdorff spaces*, Indag. Math. (N.S.)**41**(1979), 155-162. MR**535563 (80e:54026)****[10]**-,*A nonsupercompact continuous image of a supercompact space*, Houston J. Math.**5**(1979), 241-247. MR**546758 (80m:54033)****[11]**C. F. Mills,*A simpler proof that compact metric spaces are supercompact*, Proc. Amer. Math. Soc.**73**(1979), 388-390. MR**518526 (80d:54036)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1994-1209102-0

Keywords:
Supercompact,
limit,
sequence

Article copyright:
© Copyright 1994
American Mathematical Society