Products of weakly-ℵ-compact spaces

Ulmer, Milton

doi:10.1090/S0002-9947-1972-0375232-9

Products of weakly-$\aleph$-compact spaces
HTML articles powered by AMS MathViewer

by Milton Ulmer PDF

Trans. Amer. Math. Soc. 170 (1972), 279-284 Request permission

Abstract:

A space is said to be weakly- ${\aleph _1}$ -compact (or weakly-Lindelöf) provided each open cover admits a countable subfamily with dense union. We show this property in a product space is determined by finite subproducts, and by assuming that ${2^{{\aleph _0}}} = {2^{{\aleph _1}}}$ we show the property is not preserved by finite products. These results are generalized to higher cardinals and two research problems are stated.

References

Paul J. Cohen, The independence of the continuum hypothesis. II, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 105–110. MR 159745, DOI 10.1073/pnas.51.1.105
W. W. Comfort, A nonpseudocompact product space whose finite subproducts are pseudocompact, Math. Ann. 170 (1967), 41–44. MR 210070, DOI 10.1007/BF01362285
W. W. Comfort, A survey of cardinal invariants, General Topology and Appl. 1 (1971), no. 2, 163–199. MR 290326
W. W. Comfort, Neil Hindman, and S. Negrepontis, $F^{\prime }$-spaces and their product with $P$-spaces, Pacific J. Math. 28 (1969), 489–502. MR 242106
Zdeněk Frolík, Generalisations of compact and Lindelöf spaces, Czechoslovak Math. J. 9(84) (1959), 172–217 (Russian, with English summary). MR 105075
Zdeněk Frolík, Sums of ultrafilters, Bull. Amer. Math. Soc. 73 (1967), 87–91. MR 203676, DOI 10.1090/S0002-9904-1967-11653-7
Zdeněk Frolík, On two problems of W. W. Comfort, Comment. Math. Univ. Carolinae 8 (1967), 139–144. MR 210071
Anthony W. Hager, Projections of zero-sets (and the fine uniformity on a product), Trans. Amer. Math. Soc. 140 (1969), 87–94. MR 242114, DOI 10.1090/S0002-9947-1969-0242114-2
N. Noble and Milton Ulmer, Factoring functions on Cartesian products, Trans. Amer. Math. Soc. 163 (1972), 329–339. MR 288721, DOI 10.1090/S0002-9947-1972-0288721-2
Mary Ellen Rudin, Types of ultrafilters, Topology Seminar (Wisconsin, 1965) Ann. of Math. Studies, No. 60, Princeton Univ. Press, Princeton, N.J., 1966, pp. 147–151. MR 0216451
Walter Rudin, Homogeneity problems in the theory of Čech compactifications, Duke Math. J. 23 (1956), 409–419. MR 80902

Continuous functions on product spaces

The countable chain condition