CH and open subspaces of 𝐹-spaces

Dow, Alan

doi:10.1090/S0002-9939-1983-0712648-5

CH and open subspaces of $F$-spaces
HTML articles powered by AMS MathViewer

by Alan Dow

Proc. Amer. Math. Soc. 89 (1983), 341-345

DOI: https://doi.org/10.1090/S0002-9939-1983-0712648-5

PDF | Request permission

Abstract:

N. J. Fine and L. Gillman showed that, if one assumes CH, each open subset of an $F$-space of weight $c$ is an $F$-space. In this note it is shown that this fact is equivalent to CH.

References

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

-spaces embed in

Eric K. van Douwen and Jan van Mill, Parovičenko’s characterization of $\beta \omega -\omega$ implies CH, Proc. Amer. Math. Soc. 72 (1978), no. 3, 539–541. MR 509251, DOI 10.1090/S0002-9939-1978-0509251-7
Eric K. van Douwen and Jan van Mill, Subspaces of basically disconnected spaces or quotients of countably complete Boolean algebras, Trans. Amer. Math. Soc. 259 (1980), no. 1, 121–127. MR 561827, DOI 10.1090/S0002-9947-1980-0561827-0

There can be

-embedded dense proper subspaces in

Alan Dow, On $F$-spaces and $F^{\prime }$-spaces, Pacific J. Math. 108 (1983), no. 2, 275–284. MR 713737, DOI 10.2140/pjm.1983.108.275
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
Stephen H. Hechler, On a ubiquitous cardinal, Proc. Amer. Math. Soc. 52 (1975), 348–352. MR 380705, DOI 10.1090/S0002-9939-1975-0380705-7
Alain Louveau, Caractérisation des sous-espaces compacts de $\beta \textbf {N}$, Bull. Sci. Math. (2) 97 (1973), 259–263 (1974) (French). MR 353261
I. I. Parovičenko, On a universal bicompactum of weight $\aleph$, Dokl. Akad. Nauk SSSR 150 (1963), 36–39. MR 0150732
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, 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
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, A survey of absolutes of topological spaces, Topological structures, II (Proc. Sympos. Topology and Geom., Amsterdam, 1978) Math. Centre Tracts, vol. 116, Math. Centrum, Amsterdam, 1979, pp. 323–362. MR 565852