Let X be a completely
regular Hausdorff space. Denote the “absolute” (also called the “projective cover”) of
X by E(X), the Boolean algebra of regular closed subsets of X by R(X), and the
Stone-Cech compactification of X by βX. In this paper it is proved that the
canonical map k : E(βX)−≻ βX maps βE(X) − E(X) irreducibly onto βX − X if
and only if the map A → clρxA−X is a Boolean algebra homomorphism from R(X)
into R(βX −X). This latter condition is shown to hold for a wide class of spaces X.
These results are used to calculate absolutes and well-known co-absolutes of
βX−X under several different sets of hypotheses concerning the topology of
X.
