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.

For every natural number n, any continuous function on the product of X1×X2×...×Xn pseudocompact spaces extends to a separately βX1×βX2×...×βXn their Stone-Cech compactifications.

Interpreting modal diamond as the closure of a topological space, we axiomatize the modal logic of each metrizable Stone space and of each extremally disconnected Stone space. As a corollary, we obtain that S4.1 is the modal logic of the Pelczynski compactification of the natural numbers and S4.2 is the modal logic of the Gleason cover of the Cantor space. As another corollary, we obtain an axi...

In this paper, the definition of Stone-Čech defined on discrete topological space natural numbers is used, which set all ultra-filters and largest compact created from space. The algebraic properties will be used to understand study some it. We have expanded semi-group process ( , naturally compactification . use illustrate structure, emphasizing smallest ideal where show that containing a mini...