Complete bases and Wallman realcompactifications

Blasco, Jose L.

doi:10.1090/S0002-9939-1979-0529226-2

Complete bases and Wallman realcompactifications
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by Jose L. Blasco

Proc. Amer. Math. Soc. 75 (1979), 114-118

DOI: https://doi.org/10.1090/S0002-9939-1979-0529226-2

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Abstract:

We study a particular class of separating nest generated intersection rings on a Tychonoff space X, that we call complete bases. They are characterized by the equality $\beta (\upsilon (X,\mathcal {D})) = \omega (X,\mathcal {D})$ between their associated Wallman spaces. It is proven that for each separating nest generated intersection ring $\mathcal {D}$ there exists a unique complete base $\hat {\mathcal {D}}$ such that $\upsilon (X,\mathcal {D}) = \upsilon (X,\widehat {\mathcal {D}})$. From this result we obtain a necessary and sufficient condition for the existence of a continuous extension to $\upsilon (X,\mathcal {D})$ of a real-valued function over X. Some applications of these results to certain inverse-closed subalgebras of $C(X)$ are given.

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