Open and other kinds of map extensions over zero-dimensional local compactifications

Abstract

Generalizing a theorem of Ph. Dwinger (1961) [7], we describe the partially ordered set of all (up to equivalence) zero-dimensional locally compact Hausdorff extensions of a zerodimensional Hausdorff space. Using this description, we find the necessary and sufficient conditions which has to satisfy a map between two zero-dimensional Hausdorff spaces in order to have some kind of extension over arbitrary, but fixed, Hausdorff zero-dimensional local compactifications of these spaces; we consider the following kinds of extensions: continuous, open, quasi-open, skeletal, perfect, injective, surjective, dense embedding. In this way we generalize some classical results of B. Banaschewski (1955) [1] about the maximal zero-dimensional Hausdorff compactification. Extending a recent theorem of G. Bezhanishvili (2009) [2], we describe the local proximities corresponding to the zerodimensional Hausdorff local compactifications. IntroductionIn [1], B. Banaschewski proved that every zero-dimensional Hausdorff space X has a zero-dimensional Hausdorff compactification β 0 X with the following remarkable property: every continuous map f : X → Y , where Y is a zero-dimensional Hausdorff compact space, can be extended to a continuous map β 0 f : β 0 X → Y ; in particular, β 0 X is the maximal zerodimensional Hausdorff compactification of X . As far as I know, there are no descriptions of the maps f for which the extension β 0 f is open or quasi-open. In this paper we solve the following more general problem: let f : X → Y be a map between two zero-dimensional Hausdorff spaces and (lX, l X ), (lY , l Y ) be Hausdorff zero-dimensional locally compact extensions of X and Y , respectively; find the necessary and sufficient conditions which has to satisfy the map f in order to have an "extension" g : l X → lY (i.e. g • l X = l Y • f ) which is a map with some special properties (we consider the following properties: continuous, open, perfect, quasi-open, skeletal, injective, surjective, dense embedding). In [10], S. Leader solved such a problem for continuous extensions over Hausdorff local compactifications (= locally compact extensions) using the language of local proximities (which, as he showed, are in a bijective correspondence (preserving the order) with the Hausdorff local compactifications regarded up to equivalence). Hence, if one can describe the local proximities which correspond to zero-dimensional Hausdorff local compactifications then the above problem will be solved for continuous extensions. Recently, G. Bezhanishvili [2], solving an old problem of L. Esakia, described the Efremovič proximities which correspond (in the sense of the famous Smirnov Compactification Theorem [17]) to the zero-dimensional Hausdorff compactifications (and called them zero-dimensional Efremovič proximities). We extend here his result to the Leader local proximities, i.e. we describe the local proximities which correspond to the Hausdorff zero-dimensional local compactifications and call them