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An Internal Characterization of Realcompactness

Published online by Cambridge University Press:  20 November 2018

Douglas Harris
Douglas Harris*
Affiliation:
Marquette University, Milwaukee, Wisconsin
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A space is realcompact if it is a homeomorph of a closed subspace of a product of real lines. Many external characterizations of realcompactness have appeared, but there seems to be no simple internal characterization. We provide such a characterization in terms of the existence of a collection of covers of a certain type and use it to examine realcompact extensions of a space and to characterize the Q-closure of a space in a compac tification.

A structure on X is a collection of covers of X that forms a filter under refinement ordering; the members of a structure are called gauges. A balanced refinement of a gauge a is a gauge β with cardinal not greater than that of a such that for each B ∈ β there is A ∈ α such that {A, X – B} is also a gauge; thus a balanced refinement is certainly a refinement.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 3 , June 1971 , pp. 439 - 444
Copyright
Copyright © Canadian Mathematical Society 1971

References

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