Local compactness and Hewitt realcompactifications of products

Ohta, Haruto

doi:10.1090/S0002-9939-1978-0482673-9

Local compactness and Hewitt realcompactifications of products

Author: Haruto Ohta
Journal: Proc. Amer. Math. Soc. 69 (1978), 339-343
MSC: Primary 54D60
DOI: https://doi.org/10.1090/S0002-9939-1978-0482673-9
MathSciNet review: 0482673
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Abstract: In this note we prove McArthur's conjecture [6]: If card X is nonmeasurable and if $v(X \times Y) = vX \times vY$ holds for each space Y, then X is locally compact. Consequently, we can completely characterize the class of all spaces X such that for each space Y, $v(X \times Y) = vX \times vY$ holds.

[1] W. W. Comfort, On the Hewitt realcompactification of a product space, Trans. Amer. Math. Soc. 131 (1968), 107-118. MR 0222846 (36:5896)
[2] N. Dykes, Mappings and realcompact spaces, Pacific J. Math. 31 (1969), 347-358. MR 0263039 (41:7644)
[3] L. Gillman and M. Jerison, Rings of continuous functions, Van Nostrand, Princeton, N. J., 1960. MR 0116199 (22:6994)
[4] M. Hušek, Pseudo- $\mathfrak{m}$ -compactness and $v(P \times Q)$ , Indag. Math. 33 (1971), 320-326. MR 0296898 (45:5957)
[5] T. Isiwata, Topological completions and realcompactifications, Proc. Japan Acad. 47 (1971), 941-946. MR 0319151 (47:7697)
[6] W. G. McArthur, Hewitt realcompactifications of products, Canad. J. Math. 22 (1970), 645-656. MR 0266161 (42:1069)
[7] K. Morita, Topological completions and M-spaces, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 10 (1970), 271-288. MR 0271904 (42:6785)
[8] R. Pupier, Topological completion of a product, Rev. Roumaine Math. Pures Appl. 19 (1974), 925-933. MR 0362198 (50:14640)