A note on 𝒵-realcompactifications

D’Aristotle, Anthony J.

doi:10.1090/S0002-9939-1972-0288730-9

A note on $\mathcal {Z}$-realcompactifications
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by Anthony J. D’Aristotle PDF

Proc. Amer. Math. Soc. 32 (1972), 615-618 Request permission

Abstract:

Orrin Frink showed that the real-valued functions over a Tychonoff space X which may be continuously extended to $\omega (\mathcal {Z})$, the Wallman-type compactification associated with a normal base $\mathcal {Z}$ for X, are those which are $\mathcal {Z}$-uniformly continuous Let $\mathcal {Z}$ be a delta normal base on a Tychonoff space X, and let $\eta (\mathcal {Z})$ be the corresponding $\mathcal {Z}$-realcompactification of X. In this note we show that countable $\mathcal {Z}$-uniform continuity is a sufficient but not a necessary condition for continuously extending real-valued functions over X to $\eta (\mathcal {Z})$.

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