All nonPpoints are the limits of nontrivial sequences in supercompact spaces
Authors:
Zhongqiang Yang and Wei Sun
Journal:
Proc. Amer. Math. Soc. 128 (2000), 12151219
MSC (1991):
Primary 54D30
Published electronically:
August 3, 1999
MathSciNet review:
1637456
Fulltext PDF Free Access
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 nonPpoint in any continuous image of a supercompact space is the limit of a nontrivial sequence. We also prove that every nonPpoint 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. 199.
 2.
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
(93i:54015), http://dx.doi.org/10.1016/01668641(92)90005K
 3.
Eric
K. van Douwen, Special bases for compact metrizable spaces,
Fund. Math. 111 (1981), no. 3, 201–209. MR 611760
(82d:54036)
 4.
Eric
van Douwen and Jan
van Mill, Supercompact spaces, Topology Appl.
13 (1982), no. 1, 21–32. MR 637424
(82m:54017), http://dx.doi.org/10.1016/01668641(82)900049
 5.
Alan
Dow, Good and OK ultrafilters, Trans. Amer. Math. Soc. 290 (1985), no. 1, 145–160. MR 787959
(86f:54044), http://dx.doi.org/10.1090/S00029947198507879594
 6.
de Groot J., Supercompactness and superextension, Contribution to Extension Theory of Topological Structures, Proceedings 1967 Berlin Symposium, Berlin(1969), pp. 199.
 7.
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 0482661
(58 #2719)
 8.
Jan
van Mill, An introduction to 𝛽𝜔, Handbook of
settheoretic topology, NorthHolland, Amsterdam, 1984,
pp. 503–567. MR 776630
(86f:54027)
 9.
Jan
van Mill and Charles
F. Mills, Closed 𝐺_{𝛿} subsets of supercompact
Hausdorff spaces, Nederl. Akad. Wetensch. Indag. Math.
41 (1979), no. 2, 155–162. MR 535563
(80e:54026)
 10.
Edward
L. Wimmers, The Shelah 𝑃point independence theorem,
Israel J. Math. 43 (1982), no. 1, 28–48. MR 728877
(85e:03118), http://dx.doi.org/10.1007/BF02761683
 11.
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
(95a:54041), http://dx.doi.org/10.1090/S00029939199412091020
 12.
Yang Zhongqiang, A simple proof for the Alexander subbase lemma, J. Shaanxi Normal University 26(1998), pp. 199 (In Chinese).
 1.
 Bell M. G., Not all compact Hausdorff spaces are supercompact, General Topology Appl. 8(1978), pp. 199.
 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. 199. MR 93i:54015
 3.
 van Douwen E. K., Special bases for compact metrizable spaces, Fund. Math. 61(1981), pp. 201209. MR 82d:54036
 4.
 van Douwen E. K. and van Mill J., Supercompact spaces, Topology Appl. 1(1982), pp. 199. MR 82m:54017
 5.
 Dow A., Good and OK ultrafilters, Trans. Amer. Math. Soc. 290(1985), pp. 199. MR 86f:54044
 6.
 de Groot J., Supercompactness and superextension, Contribution to Extension Theory of Topological Structures, Proceedings 1967 Berlin Symposium, Berlin(1969), pp. 199.
 7.
 van Mill J., A countable space no compactification of which is supercompact, Bull. Acad. Pol. Sci. 25(1972), pp. 199. MR 58:2719
 8.
 van Mill J., A introduction to , Handbook of SetTheoretic Topology (ed. by Kunen K. and Vaughan J.E.), pp. 503568. MR 86f:54027
 9.
 van Mill J. and Mills C. F., Closed subsets of supercompact Hausdorff spaces, Indag. Math. (N.S.) 41(1979), pp. 155162. MR 80e:54026
 10.
 Wimmers E. L., The Shelah Ppoint independence theorem, Israel J. Math. 43(1982), pp. 199. 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. 199. MR 95a:54041
 12.
 Yang Zhongqiang, A simple proof for the Alexander subbase lemma, J. Shaanxi Normal University 26(1998), pp. 199 (In Chinese).
Similar Articles
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:
http://dx.doi.org/10.1090/S0002993999051199
PII:
S 00029939(99)051199
Keywords:
Supercompact,
Ppoint,
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
