A topological space X is said to be completely regular if whenever F is a nonempty closed set and x ∈ X \F , there is a continuous function f : X → [0, 1] such that f(x) = 0 and f(F ) = {1}. A completely regular space need not be Hausdorff. For example, ifX is any set with more than one point, then the trivial topology, in which the only closed sets are ∅ and X, is vacuously completely regular,...

J. Keesling has shown that for connected spaces X the natural inclusion e : X → βX of X in its Stone-Čech compactification is a shape equivalence if and only if X is pseudocompact. This paper establishes the analogous result for strong shape. Moreover, pseudocompact spaces are characterized as spaces which admit compact resolutions, which improves a result of I. Lončar.

The families of right (left) translation finite subsets of a discrete infinite group Γ are defined and shown to be ideals. Their kernels ZR and ZL are identified as the closure of the set of products pq (p · q) in the Čech-Stone compactification βΓ. Consequently it is shown that the map π : βΓ → Γ , the canonical semigroup homomorphism from βΓ onto Γ , the universal semitopological semigroup co...