## On first countable, countably compact spaces. I. $(\omega _{1}, \omega ^{\ast } _{1})$-gaps

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- 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
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## 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**
—, - Ryszard Engelking,
*Topologia ogólna*, Biblioteka Matematyczna [Mathematics Library], vol. 47, Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1975 (Polish). MR**0500779**
F. Hausdorff, - 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
—, - Stephen Willard,
*General topology*, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR**0264581**

*Functions from the integers to the integers and topology*, cocktail napkin, Hilton Hotel Bar, San Antonio, Texas, 1975.

*Summen von*${\aleph _1}$

*mengen*, Fund. Math.

**26**(1936), 241-255.

*Some cardinals related to*$c$

*and topology*(Topology Conf., 1979), Metric Spaces, Generalized Metric Spaces, Continua, Math. Dept., Guilford College, Greensboro.

## Bibliographic Information

- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**279**(1983), 463-469 - MSC: Primary 54A35; Secondary 03E35, 03E50, 54D15
- DOI: https://doi.org/10.1090/S0002-9947-1983-0709563-4
- MathSciNet review: 709563