Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Fundamenta Mathematicae / All issues

Abstract

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 $F(X)/C_{\rm K}(X)$ and the lattice of all compactifications of $X$ with zero-dimensional remainder, the finite-dimensional subalgebras corresponding to the compactifications with finite remainder.