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A characterization of realcompactness. (English) Zbl 0860.54025

For a Tikhonov space \(X\), let \(Z(X)\) be the collection of all zero-sets in \(X\). The author defines a space \(X\) to be \(z\)-realcompact if whenever \({\mathcal F}\) is an ultrafilter of cozero-sets of \(X\) such that \(\{Z\in Z(X): (\exists F\in {\mathcal F}) (F\subseteq Z)\}\) has the countable intersection property, then \(\bigcap \{\overline F: F\in {\mathcal F}\} \neq\emptyset\). Main results are: (1) A Tikhonov space \(X\) is realcompact if and only if \(X\) is \(z\)-realcompact. (2) Let \(\gamma\) be the collection of all countable cozero covers of \(X\). Then \(X\) is realcompact if and only if \(\gamma\) is complete.
{Reviewer’s remark: Essentially the same result as (2) is known. T. Shirota [Osaka Math. J. 4, 23-40 (1952; Zbl 0047.41704)] proved that a space \(X\) is realcompact if and only if the collection of all countable normal covers of \(X\) is complete. It is known that a countable cozero cover is normal and, conversely, a countable normal cover has a countable cozero refinement}.

MSC:

54D60 Realcompactness and realcompactification

Citations:

Zbl 0047.41704
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References:

[1] Z. FrolĂ­k, A generalization of realcompact spaces, Czech. Math. J., 13 (1963), 127-138. · Zbl 0112.37603
[2] L. Gillman and M. Jerison, Rings of Continuous Functions, University Series in Higher Math, Van Nostrand (Princeton, New Jersey, 1960). · Zbl 0093.30001
[3] M. D. Weir, Hewitt?Nachbin Spaces, North Holland Math. Studies, American Elsevier (New York, 1975). · Zbl 0314.54002
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