On first countable, countably compact spaces. I. (𝜔₁,𝜔*₁)-gaps

Nyikos, Peter J.; Vaughan, Jerry E.

doi:10.1090/S0002-9947-1983-0709563-4

On first countable, countably compact spaces. I. $(\omega _{1}, \omega ^{\ast } _{1})$-gaps
HTML articles powered by AMS MathViewer

by Peter J. Nyikos and Jerry E. Vaughan

Trans. Amer. Math. Soc. 279 (1983), 463-469

DOI: https://doi.org/10.1090/S0002-9947-1983-0709563-4

PDF | Request permission

Abstract:

This paper is concerned with the $({\omega _1},\omega _1^{\ast })$-gaps of F. Hausdorff and the topological spaces defined from them by Eric van Douwen. We construct special gaps in order that the associated gap spaces will have interesting topological properties. For example, the gap spaces we construct show that in certain models of set theory, there exist countably compact, first countable, separable, nonnormal ${T_2}$-spaces.

References

David Booth, Ultrafilters on a countable set, Ann. Math. Logic 2 (1970/71), no. 1, 1–24. MR 277371, DOI 10.1016/0003-4843(70)90005-7
Aleksander Błaszczyk and Andrzej Szymański, Hausdorff’s gaps versus normality, Bull. Acad. Polon. Sci. Sér. Sci. Math. 30 (1982), no. 7-8, 371–378 (English, with Russian summary). MR 707751
Eric K. van Douwen, Hausdorff gaps and a nice countably paracompact nonnormal space, Topology Proceedings, Vol. I (Conf., Auburn Univ., Auburn, Ala., 1976) Math. Dept., Auburn Univ., Auburn, Ala., 1977, pp. 239–242. MR 0454915

Functions from the integers to the integers and topology

Ryszard Engelking, Topologia ogólna, Biblioteka Matematyczna [Mathematics Library], vol. 47, Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1975 (Polish). MR 0500779

Summen von

mengen

26

Stephen H. Hechler, A dozen small uncountable cardinals, TOPO 72—general topology and its applications (Proc. Second Pittsburgh Internat. Conf., Pittsburgh, Pa., 1972; dedicated to the memory of Johannes H. de Groot), Lecture Notes in Math., Vol. 378, Springer, Berlin, 1974, pp. 207–218. MR 0369078
Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
Peter J. Nyikos, The topological structure of the tangent and cotangent bundles on the long line, Topology Proc. 4 (1979), no. 1, 271–276 (1980). MR 583709
A. J. Ostaszewski, On countably compact, perfectly normal spaces, J. London Math. Soc. (2) 14 (1976), no. 3, 505–516. MR 438292, DOI 10.1112/jlms/s2-14.3.505
Fritz Rothberger, On some problems of Hausdorff and of Sierpiński, Fund. Math. 35 (1948), 29–46. MR 29958, DOI 10.4064/fm-35-1-29-46
J. E. Vaughan, A countably compact, first countable, nonnormal $T_{2}$-space, Proc. Amer. Math. Soc. 75 (1979), no. 2, 339–342. MR 532163, DOI 10.1090/S0002-9939-1979-0532163-0

Some cardinals related to

and topology

Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0264581