On nowhere dense closed 𝑃-sets

Kunen, Kenneth; van Mill, Jan; Mills, Charles F.

doi:10.1090/S0002-9939-1980-0548097-X

On nowhere dense closed $P$-sets
HTML articles powered by AMS MathViewer

by Kenneth Kunen, Jan van Mill and Charles F. Mills PDF

Proc. Amer. Math. Soc. 78 (1980), 119-123 Request permission

Abstract:

We show that no compact space of weight ${\omega _1}$ can be covered by nowhere dense closed P-sets. In addition, we construct a compact space of weight ${\omega _2}$ which can be covered by nowhere dense closed P-sets. As an application, we show that CH is equivalent to the statement that each small nonpseudocompact space has a remote point.

References

Remote points

Spaces without remote points

N. J. Fine and L. Gillman, Remote points in $\beta R$, Proc. Amer. Math. Soc. 13 (1962), 29–36. MR 143172, DOI 10.1090/S0002-9939-1962-0143172-5
Leonard Gillman and Melvin Henriksen, Concerning rings of continuous functions, Trans. Amer. Math. Soc. 77 (1954), 340–362. MR 63646, DOI 10.1090/S0002-9947-1954-0063646-5
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
Jan van Mill and Charles F. Mills, A topological property enjoyed by near points but not by large points, Topology Appl. 11 (1980), no. 2, 199–209. MR 572374, DOI 10.1016/0166-8641(80)90008-5
R. Grant Woods, Co-absolutes of remainders of Stone-Čech compactifications, Pacific J. Math. 37 (1971), 545–560. MR 307179, DOI 10.2140/pjm.1971.37.545