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Extra Countably Compact Spaces

Published online by Cambridge University Press:  20 November 2018

Victor Saks
Victor Saks*
Affiliation:
Daemen College, AmherstNew York 14226
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Abstract

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A completely regular HausdorfT space is extra countably compact if every infinite subset of βX has an accumulation point in X. It is a theorem of Comfort and Waiveris that if X either an F-space or realcompact (topologically complete), then there is a set {Pξ:ξ<2C} of extra countably compact (countably compact) subspaces of αX such that Pξ ∩ Pξ = X, for ξ<ξ'<2C. Comfort and Waiveris conjecture that in all three cases, the spaces may be chosen pairwise non-homeomorphic. We prove this conjecture, using D- limits and weak P-points. We also give a partial solution to another question asked by Comfort and Waiveris.

Keywords

54D3054D3554A25Key words and phrasesExtra countably compacthomeomorphicD-limitsweak P-pointscountably compact
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 26 , Issue 4 , 01 December 1983 , pp. 473 - 481
Copyright
Copyright © Canadian Mathematical Society 1983

References

[B] Bernstein, A. R., A new kind of compactness for topological spaces, Fund. Math. 66 (1970), 185-193.Google Scholar
[BvD] Burke, D. K. and van Douwen, E. K., On countably compact extensions of normal locally compact M-spaces, in Set-theoretic topology, Academic Press, New York, 1977, 81-89.Google Scholar
[CN] Comfort, W. W. and Negrepontis, S., The theory of ultrafilters, Grundlehren der math. Wissenschaften vol. 211, Springer-Verlag, New York-Heidelberg-Berlin, 1974.Google Scholar
[CW] Comfort, W. W. and Waiveris, C., Intersections of countably compact subspaces of Stone-Čech compactifications, Uspehi Mat. Nauk 35 (1980), 67-77 [in Russian] (English translation: Russian Mathematical Surveys).Google Scholar
[vD] Van Douwen, E. K., The product of two countably compact topological groups, Trans. AMS 262 No. 2, (1980), 417-427.Google Scholar
[vDF] van Douwen, E. K. and Fleissner, W. G., The definable forcing axiom, preprint.Google Scholar
[Fe] Fedorčuk, V., A compact space of cardinality the continuum with no convergent sequences, Proc. Camb. Phil. Soc. 81 (1977), 177-181.Google Scholar
[Fk] Froliќ, Z., Sums of ultrafilters, Bull. AMS, 73 (1967), 87-91.Google Scholar
[Fr] Franklin, S. P., On two questions of Moore and Mrowka, Proc. AMS 21 (1969), 597-599.Google Scholar
[GJ] Gillman, L. and Jerison, M., Rings of continuous functions, D. van Nostrand Co., Princeton, N.J., 1960.Google Scholar
[GS] Ginsburg, J. and Saks, V., Some applications of ultrafilters in topology, Pacific J. Math. 57 (1975), 403-418.Google Scholar
[Ka] Kato, A., Various countably-compactifications and their applications, General Topology and Appl. 8 (1978) 24-46.Google Scholar
[Kj] Kunen, K., Some Points In |3n, Proc. Canm. Phil. Soc. 80 (1976), 385-398.Google Scholar
[K2] Kunen, K., Weak P-points in N*, Proc. Bolyai Janos Soc. Coll. on Top., Budapest (1978).Google Scholar
[K3] Kunen, K., Inhomogeneity of products of compact F-spaces, Handwritten notes, privately circulated.Google Scholar
[MS] Malyhin, V. I. and Šapirovskiì, B. E., Martin's Axiom and properties of topological spaces, Dokl. Ak. Nauk SSSR 213 (1973) = Sov. Math. Dok. 14 (1973), 1746-1751.Google Scholar
[R] Rudin, W., Homogeneity problems in the theory of Čech compactifications, Duke Math. J. 23 (1956), 409-420, 633.Google Scholar
[SJ] Saks, V., Ultrafilter invariants in topological spaces. Trans. AMS 241 (1978), 19–91.Google Scholar
[S2] Saks, V., Products of countably compact spaces, Top. Proc. 4 (1979), 553-575.Google Scholar
[ST] Szymański, A. and Turzański, M., αN and sequential compactness, Coll. Math. 35 (1976), 205-208.Google Scholar
[VJ] Vaughan, J. E., A countably compact space and its products, Proc. AMS 71 (1978), 133-137.Google Scholar
[V2] Vaughan, J.E., Discrete sequences of points, Top. Proc. 3 No. 1, (1978), 237-265.Google Scholar
[W] Waiveris, C., Intersections of countably compact subspaces of αX, Top Proc. 4 (1979), 177-191.Google Scholar
[Wo] Woods, R. G., Some ℵ0-bounded subsets of Stone-Čech compactifications Israel J. Math. 9 No. 2, (1971), 250-256.Google Scholar