All non-P-points are the limits of nontrivial sequences in supercompact spaces

Authors:
Zhongqiang Yang and Wei Sun

Journal:
Proc. Amer. Math. Soc. **128** (2000), 1215-1219

MSC (1991):
Primary 54D30

DOI:
https://doi.org/10.1090/S0002-9939-99-05119-9

Published electronically:
August 3, 1999

MathSciNet review:
1637456

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 -subspace of a supercompact space is a cluster point of a subset with cardinal number But we do not know whether this statement holds when replacing by the countable cardinal number. As an application, we prove in ZFC that there exists a countable stratifiable space which has no supercompact compactification.

**1.**Bell M. G.,*Not all compact Hausdorff spaces are supercompact,*General Topology Appl.**8**(1978), pp. 1-99.**2.**Bula W., Nikiel J., Tuncali H. M. and Tymchatyn E.D.,*Continuous images of ordered compacta are regular supercompact,*Topology Appl.**45**(1992), pp. 1-99. MR**93i:54015****3.**van Douwen E. K.,*Special bases for compact metrizable spaces,*Fund. Math.**61**(1981), pp. 201-209. MR**82d:54036****4.**van Douwen E. K. and van Mill J.,*Supercompact spaces,*Topology Appl.**1**(1982), pp. 1-99. MR**82m:54017****5.**Dow A.,*Good and OK ultrafilters,*Trans. Amer. Math. Soc.**290**(1985), pp. 1-99. MR**86f:54044****6.**de Groot J.,*Supercompactness and superextension,*Contribution to Extension Theory of Topological Structures, Proceedings 1967 Berlin Symposium, Berlin(1969), pp. 1-99.**7.**van Mill J.,*A countable space no compactification of which is supercompact,*Bull. Acad. Pol. Sci.**25**(1972), pp. 1-99. MR**58:2719****8.**van Mill J.,*A introduction to ,*Handbook of Set-Theoretic Topology (ed. by Kunen K. and Vaughan J.E.), pp. 503-568. MR**86f:54027****9.**van Mill J. and Mills C. F.,*Closed subsets of supercompact Hausdorff spaces,*Indag. Math. (N.S.)**41**(1979), pp. 155-162. MR**80e:54026****10.**Wimmers E. L.,*The Shelah P-point independence theorem,*Israel J. Math.**43**(1982), pp. 1-99. MR**85e:03118****11.**Yang Zhongqiang,*All cluster points of countable sets in supercompact spaces are the limits of nontrivial sequences,*Proc. Amer. Math. Soc.**122**(1994), pp. 1-99. MR**95a:54041****12.**Yang Zhongqiang,*A simple proof for the Alexander subbase lemma,*J. Shaanxi Normal University**26**(1998), pp. 1-99 (In Chinese).

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
54D30

Retrieve articles in all journals with MSC (1991): 54D30

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

DOI:
https://doi.org/10.1090/S0002-9939-99-05119-9

Keywords:
Supercompact,
P-point,
sequence,
compactification

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

Article copyright:
© Copyright 2000
American Mathematical Society