For X a Tikhonov space, let F(X) be the algebra of all real-valued continuous functions on X that assume only finitely many values outside some compact subset. We show that F(X) generates a compactification γX of X if and only if X has a base of open sets whose boundaries have compact neighborhoods, and we note that if this happens then γX is the Freudenthal compactification of X. For X Hausdorff and locally compact, we establish an isomorphism between the lattice of all subalgebras of and the lattice of all compactifications of X with zero-dimensional remainder, the finite-dimensional subalgebras corresponding to the compactifications with finite remainder.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm178-3-2,
author = {Jes\'us M. Dom\'\i nguez},
title = {A generating family for the Freudenthal compactification of a class of rimcompact spaces},
journal = {Fundamenta Mathematicae},
volume = {177},
year = {2003},
pages = {203-215},
zbl = {1054.54021},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm178-3-2}
}
Jesús M. Domínguez. A generating family for the Freudenthal compactification of a class of rimcompact spaces. Fundamenta Mathematicae, Tome 177 (2003) pp. 203-215. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm178-3-2/