All non-$\text {P}$-points are the limits of nontrivial sequences in supercompact spaces
HTML articles powered by AMS MathViewer
- by Zhongqiang Yang and Wei Sun PDF
- Proc. Amer. Math. Soc. 128 (2000), 1215-1219 Request permission
Abstract:
A Hausdorff topological space is called supercompact if there exists a subbase such that every cover consisting of this subbase has a subcover consisting of two elements. In this paper, we prove that every non-P-point in any continuous image of a supercompact space is the limit of a nontrivial sequence. We also prove that every non-P-point in a closed $G_{\delta }$-subspace of a supercompact space is a cluster point of a subset with cardinal number $\leq c.$ But we do not know whether this statement holds when replacing $c$ by the countable cardinal number. As an application, we prove in ZFC that there exists a countable stratifiable space which has no supercompact compactification.References
- Bell M. G., Not all compact Hausdorff spaces are supercompact, General Topology Appl. 8(1978), pp. 151-155.
- 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
- Alan Dow, Good and OK ultrafilters, Trans. Amer. Math. Soc. 290 (1985), no. 1, 145–160. MR 787959, DOI 10.1090/S0002-9947-1985-0787959-4
- de Groot J., Supercompactness and superextension, Contribution to Extension Theory of Topological Structures, Proceedings 1967 Berlin Symposium, Berlin(1969), pp. 89-90.
- J. van Mill, A countable space no compactification of which is supercompact, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 11, 1129–1132 (English, with Russian summary). MR 482661
- Jan van Mill, An introduction to $\beta \omega$, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 503–567. MR 776630
- 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, DOI 10.1016/S1385-7258(79)80010-4
- Edward L. Wimmers, The Shelah $P$-point independence theorem, Israel J. Math. 43 (1982), no. 1, 28–48. MR 728877, DOI 10.1007/BF02761683
- Zhong Qiang Yang, All cluster points of countable sets in supercompact spaces are the limits of nontrivial sequences, Proc. Amer. Math. Soc. 122 (1994), no. 2, 591–595. MR 1209102, DOI 10.1090/S0002-9939-1994-1209102-0
- Yang Zhongqiang, A simple proof for the Alexander subbase lemma, J. Shaanxi Normal University 26(1998), pp. 23-26 (In Chinese).
Additional Information
- Zhongqiang Yang
- Affiliation: Department of Mathematics, Shaanxi Normal University, Xi’an, 710062, People’s Republic of China
- Email: yangmathsnuc@ihw.com.cn
- Wei Sun
- Affiliation: Xi’an Institute of Technology, Xi’an, 710032, People’s Republic of China
- Received by editor(s): March 8, 1998
- Received by editor(s) in revised form: May 20, 1998
- Published electronically: August 3, 1999
- Additional Notes: This work is supported by the National Education Committee of China for outstanding youths and by the National Education Committee of China for Scholars returning from abroad.
- Communicated by: Alan Dow
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 1215-1219
- MSC (1991): Primary 54D30
- DOI: https://doi.org/10.1090/S0002-9939-99-05119-9
- MathSciNet review: 1637456