Two extensions of the stone duality to the category of zero-dimensional Hausdorff spaces

Extending the Stone Duality Theorem, we prove two duality theorems for the category ZHaus of zero-dimensional Hausdorff spaces and continuous maps. They extend also the Tarski Duality Theorem; the latter is even derived from one of them. We prove as well two new duality theorems for the category EDTych of extremally disconnected Tychonoff spaces and continuous maps. Also, we describe two categories which are dually equivalent to the category ZComp of zero-dimensional Hausdorff compactifications of zero-dimensional Hausdorff spaces and obtain as a corollary the Dwinger Theorem about zero-dimensional compactifications of a zero-dimensional Hausdorff space.

A CONSTRUCTION FOR SEMI-COMPACTIFICATIONS

Some new concepts of semi-compact spaces and semi-compactifications are introduced. An axiomatic construction for all Hausdorff compactifications is extended to construct semi-compactifications.

A CONSTRUCTION FOR TOPOLOGICAL EXTENSIONS

A construction for all Hausdorff compactifications given in the article [2] is analysed further to obtain other topological extensions, namely, regular extensions and normal extensions. The method is also applied to derive and to study convex compactifications.

Magill-type theorems for mappings

Magill's and Rayburn's theorems on the homeomorphism of Stone-Čech remainders and some of their generalizations to the remainders of arbitrary Hausdorff compactifications of Tychonoff spaces are extended to some class of mappings.

A quasi-uniform characterization of Wallman type compactifications

A *-compactification of a T1 quasi-uniform space (X,U) is a compact T1 quasi-uniform space (Y,V) that has a T(V*)-dense subspace quasi-isomorphic to (X,U), where V* denotes the coarsest uniformity finer than V.In this paper we characterize all Wallman type compactifications of a T1 topological space in terms of the *-compactification of its point symmetric totally bounded transitive compatible quasi-uniformities. We deduce that the *-compactification of the Pervin quasi-uniformity of any normal T1 topological space X is exactly the Stone-Cech compactification of X. We also obtain a characterization of those Hausdorff compactifications of a given space, which are of Wallman type.

Hausdorff compactifications and zero-one measures II

<p>The notion of PBS-sublattice is introduced and, using it, a simplification of the results of [6] and of some results of [5] is obtained. Two propositions concerning Wallman-type compactifications are presented as well.</p>