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Products of nearly compact spaces

1966; American Mathematical Society; Volume: 124; Issue: 1 Linguagem: Inglês

10.1090/s0002-9947-1966-0203679-7

1088-6850

C. T. Scarborough, A. H. Stone,

Advanced Banach Space Theory

A. H. STONE(i) 1. Introduction.Topological properties similar to, but slightly weaker than, compactness have often been considered (see [22, pp.901, 902] for a partial list) and it would be desirable to know to what extent the analog of Tychonoff's theorem (that every product of compact spaces is compact) will hold for them.That it is liable not to hold in general is shown by J. Novak's example [17], in which the product of two countably compact regular spaces is so far from compact as to have an open-closed infinite discrete subspace.On the other hand, Chevalley and Frink [10] have shown that every product of absolutely closed spaces is absolutely closed; Ikenaga [15] has proved a similar theorem for minimal Hausdorff spaces(2); and Glicksberg [11] has demonstrated that the product theorem holds for pseudocompact spaces under simple supplementary hypotheses (though not without them).In this paper, we first ( §2) extend the known results about absolutely closed and minimal Hausdorff spaces, reformulating them so that they apply to non-Hausdorff spaces.For minimal regular spaces the question is still open ; however, we prove some results in this direction in §3.We obtain partial results for products of feebly compact spaces in §4 (our results here are slight generalizations of those of Glicksberg [11]), and for products of sequentially compact spaces in §5(3).We use the term "space" to mean "topological space", no separation axioms being assumed unless they are stated explicitly."Compact" thus does not imply "Hausdorff", but coincides with Bourbaki's "quasicompact".We denote the space (X,3~) by X, leaving the topology 9~ to be understood.If {Xa | a e A} is a family of spaces, we write their product X = n{^|a e Á) as \~[Xa for short;it is understood that X is given the usual product topology, and (to eliminate Presented to the Society, January 24, 1964, in part, under the title The product theorem for minimal Hausdorff spaces and related results; received by the editors September 14, 1965.