All cluster points of countable sets in supercompact spaces are the limits of nontrivial sequences
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- by Zhong Qiang Yang PDF
- Proc. Amer. Math. Soc. 122 (1994), 591-595 Request permission
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 ${G_\delta }$-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.References
- Murray G. Bell, Supercompactness of compactifications and hyperspaces, Trans. Amer. Math. Soc. 281 (1984), no. 2, 717–724. MR 722770, DOI 10.1090/S0002-9947-1984-0722770-0
- Murray G. Bell, Not all dyadic spaces are supercompact, Comment. Math. Univ. Carolin. 31 (1990), no. 4, 775–779. MR 1091375
- Murray G. Bell, A first countable supercompact Hausdorff space with a closed $G_{\delta }$ nonsupercompact subspace, Colloq. Math. 43 (1980), no. 2, 233–241 (1981). MR 628178, DOI 10.4064/cm-43-2-233-241
- W. Bula, J. Nikiel, H. M. Tuncali, and E. D. Tymchatyn, Continuous images of ordered compacta are regular supercompact, Proceedings of the Tsukuba Topology Symposium (Tsukuba, 1990), 1992, pp. 203–221. MR 1180810, DOI 10.1016/0166-8641(92)90005-K
- Eric K. van Douwen, Special bases for compact metrizable spaces, Fund. Math. 111 (1981), no. 3, 201–209. MR 611760, DOI 10.4064/fm-111-3-201-209
- Eric van Douwen and Jan van Mill, Supercompact spaces, Topology Appl. 13 (1982), no. 1, 21–32. MR 637424, DOI 10.1016/0166-8641(82)90004-9 J. de Groot, Supercompactness and superextension, Contribution to Ext. Theory Top. Struct. Symposium (Berlin), Dentyscher Verlaae der Wissenchafen, Berlin, 1969, pp. 89-90.
- J. van Mill, Supercompactness and Wallman spaces, Mathematical Centre Tracts, No. 85, Mathematisch Centrum, Amsterdam, 1977. MR 0464160
- Jan van Mill and Charles F. Mills, Closed $G_{\delta }$ subsets of supercompact Hausdorff spaces, Nederl. Akad. Wetensch. Indag. Math. 41 (1979), no. 2, 155–162. MR 535563
- Charles F. Mills and Jan van Mill, A nonsupercompact continuous image of a supercompact space, Houston J. Math. 5 (1979), no. 2, 241–247. MR 546758
- Charles F. Mills, A simpler proof that compact metric spaces are supercompact, Proc. Amer. Math. Soc. 73 (1979), no. 3, 388–390. MR 518526, DOI 10.1090/S0002-9939-1979-0518526-8
Additional Information
- © Copyright 1994 American Mathematical Society
- 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