Compactness and compactifications in generalized topology

A generalized topology in a set X   is a collection CovXCovX of families of subsets of X   such that the triple (X,⋃CovX,CovX)(X,⋃CovX,CovX) is a generalized topological space (a gts) in the sense of Delfs and Knebusch. In this work, a new notion of a strict compactification of a generalized topological space is introduced and investigated in ZF. The Ultrafilter Theorem (in abbreviation UFT) is shown to be equivalent to the compactness of every Wallman extension of an arbitrary semi-normal space. That every small weakly normal gts has its Wallman strict compactification as well as several other sentences are proved to be equivalent to UFT. Among results concerning categories, it is shown that the construct of partially topological gtses is topological in ZF. If this is needed, the axiom of the existence of a universe is assumed. Many illuminating examples are given and open problems are posed.