## 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

### Keywords:

$$z$$-realcompact

Zbl 0047.41704
Full Text:

### References:

 [1] Z. Frolík, 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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