A generalization of 𝐹-spaces and some topological characterizations of GCH

Swardson, Mary Anne

doi:10.1090/S0002-9947-1983-0709575-0

A generalization of $F$-spaces and some topological characterizations of GCH
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Trans. Amer. Math. Soc. 279 (1983), 661-675 Request permission

Abstract:

Several topological characterizations involving $F$-spaces of the continuum hypothesis are due to R. G Woods and E. K. van Douwen. We extend this work by defining a space $X$ to be an ${F_\alpha }$-space if the union of $< \alpha$ cozero-sets is ${C^{\ast }}$-embedded in $X$ and by giving, for every infinite cardinal $\alpha$, topological characterizations involving ${F_\alpha }$-spaces of the cardinal equality ${2^\alpha } = {\alpha ^ + }$ .

References

W. W. Comfort and Anthony W. Hager, Estimates for the number of real-valued continuous functions, Trans. Amer. Math. Soc. 150 (1970), 619–631. MR 263016, DOI 10.1090/S0002-9947-1970-0263016-X
W. W. Comfort and S. Negrepontis, The theory of ultrafilters, Die Grundlehren der mathematischen Wissenschaften, Band 211, Springer-Verlag, New York-Heidelberg, 1974. MR 0396267, DOI 10.1007/978-3-642-65780-1
H. H. Corson, Normality in subsets of product spaces, Amer. J. Math. 81 (1959), 785–796. MR 107222, DOI 10.2307/2372929
Eric K. van Douwen, A basically disconnected normal space $\Phi$ with $\beta \Phi -\Phi =1$, Canadian J. Math. 31 (1979), no. 5, 911–914. MR 546947, DOI 10.4153/CJM-1979-086-3
Eric K. van Douwen, Remote points, Dissertationes Math. (Rozprawy Mat.) 188 (1981), 45. MR 627526
Eric K. van Douwen and Hao Xuan Zhou, The number of cozero-sets is an $\omega$-power, Topology Appl. 33 (1989), no. 2, 115–126. MR 1020274, DOI 10.1016/S0166-8641(89)80001-X

On

spaces and

spaces

Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
N. J. Fine and L. Gillman, Extension of continuous functions in $\beta N$, Bull. Amer. Math. Soc. 66 (1960), 376–381. MR 123291, DOI 10.1090/S0002-9904-1960-10460-0
Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199, DOI 10.1007/978-1-4615-7819-2

Summen von

Mengen

26

J. F. Kennison, $m$-pseudocompactness, Trans. Amer. Math. Soc. 104 (1962), 436–442. MR 145478, DOI 10.1090/S0002-9947-1962-0145478-7
J. Donald Monk, Introduction to set theory, McGraw-Hill Book Co., New York-London-Sydney, 1969. MR 0286668
Charles W. Neville and Stuart P. Lloyd, $\aleph$-projective spaces, Illinois J. Math. 25 (1981), no. 1, 159–168. MR 602906
Mary Anne Swardson, The character of certain closed sets, Canad. J. Math. 36 (1984), no. 1, 38–57. MR 733706, DOI 10.4153/CJM-1984-004-1

Generalizations of

spaces and some topological characterizations of the generalized continuum hypothesis

Mary Anne Swardson, Some topological characterizations of the generalized continuum hypothesis, Rings of continuous functions (Cincinnati, Ohio, 1982) Lecture Notes in Pure and Appl. Math., vol. 95, Dekker, New York, 1985, pp. 283–287. MR 789278
Toshiji Terada, On remote points in $\upsilon X-X$, Proc. Amer. Math. Soc. 77 (1979), no. 2, 264–266. MR 542095, DOI 10.1090/S0002-9939-1979-0542095-X
Russell C. Walker, The Stone-Čech compactification, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 83, Springer-Verlag, New York-Berlin, 1974. MR 0380698, DOI 10.1007/978-3-642-61935-9
R. Grant Woods, Characterizations of some $C^*$-embedded subspaces of $\beta \b N$, Pacific J. Math. 65 (1976), no. 2, 573–579. MR 425905, DOI 10.2140/pjm.1976.65.573
R. Grant Woods, The structure of small normal $F$-spaces, Topology Proceedings, Vol. I (Conf., Auburn Univ., Auburn, Ala., 1976) Math. Dept., Auburn Univ., Auburn, Ala., 1977, pp. 173–179. MR 0454921