A generating family for the Freudenthal compactification of a class of rimcompact spaces

Jesús M. Domínguez¹

¹ Departamento de Álgebra, Geometría y Topología Facultad de Ciencias Universidad de Valladolid 47005 Valladolid, Spain

Fundamenta Mathematicae, Tome 178 (2003) no. 3, pp. 203-215.

Résumé

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 $\gamma X$ 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 $\gamma X$ is the Freudenthal compactification of $X$. For $X$ Hausdorff and locally compact, we establish an isomorphism between the lattice of all subalgebras of $F(X)/C_{\mathrm{K}}(X)$ and the lattice of all compactifications of $X$ with zero-dimensional remainder, the finite-dimensional subalgebras corresponding to the compactifications with finite remainder.

Détail

DOI : 10.4064/fm178-3-2

Mots-clés : tikhonov space algebra real valued continuous functions assume only finitely many values outside compact subset generates compactification gamma only has base sets whose boundaries have compact neighborhoods note happens gamma freudenthal compactification hausdorff locally compact establish isomorphism between lattice subalgebras lattice compactifications zero dimensional remainder finite dimensional subalgebras corresponding compactifications finite remainder