Given a pair of spaces X and
Y , a necessary and sufficient condition is found for Y to be homeomorphic to cl
αX(αX −X) for some compactification αX of X. From this follows a necessary and
sufficient condition for Y to be homeomorphic to αX − X for some αX. As an
application, a sufficient condition is found to insure the isomorphism of the upper
semi-lattices of compactifications K(X) and K(Y ) for arbitrary X and Y , and in
consequence it appears that for every space X, there is a pseudocompact space Y
with K(X) isomorphic to K(Y ). A necessary condition for K(X) to be
isomorphic to K(Y ) is observed for arbitrary X and Y , and this leads to
the consideration of spaces compactly generated at infinity. Examples are
constructed.
