A Boolean algebra of regular closed subsets of $\beta X-X$

Abstract:

Let X be a locally compact, $\sigma$-compact, noncompact Hausdorff space. Let $\beta X$ denote the Stone-Čech compactification of X. Let $R(X)$ denote the Boolean algebra of all regular closed subsets of the topological space X. We show that the map $A \to ({\text {cl}_{\beta X}}A) - X$ is a Boolean algebra homomorphism from $R(X)$ into $R(\beta X - X)$. Assuming the continuum hypothesis, we show that if X has no more than ${2^{{\aleph _0}}}$ zero-sets, then the image of a certain dense subalgebra of $R(X)$ under this homomorphism is isomorphic to the Boolean algebra of all open-and-closed subsets of $\beta N - N$ (N denotes the countable discrete space). As a corollary, we show that there is a continuous irreducible mapping from $\beta N - N$ onto $\beta X - X$. Some theorems on higher-cardinality analogues of Baire spaces are proved, and these theorems are combined with the previous result to show that if S is a locally compact, $\sigma$-compact noncompact metric space without isolated points, then the set of remote points of $\beta S$ (i.e. those points of $\beta S$ that are not in the $\beta S$-closure of any discrete subspace of S) can be embedded densely in $\beta N - N$.