Wallman-type compactifications and products

Kost, Frank

doi:10.1090/S0002-9939-1971-0281159-8

Wallman-type compactifications and products
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by Frank Kost PDF

Proc. Amer. Math. Soc. 29 (1971), 607-612 Request permission

Abstract:

Y is a Wallman-type compactification (O. Frink, Amer. J. Math. 86 (1964), 602-607) of X in case there is a normal base Z for the closed sets of X such that the ultrafilter space from Z, denoted $\omega (Z)$, is topologically Y. It is not known if every compactification is Wallman-type. For ${Z_\alpha }$ a normal base for the closed sets of ${X_\alpha }$ for each a belonging to an index set $\Delta$ it is shown that the Tychonoff product space ${\prod _{\alpha \in \Delta }}\omega ({Z_\alpha })$ is a Wallman compactification of ${\prod _{\alpha \in \Delta }}{X_\alpha }$. Also for $X \subset T \subset \omega (Z)$ with Z a normal base for the closed sets of X, a proof that $\omega (Z)$ is a Wallman-type compactification of T is indicated.

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