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

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 zero-dimensional 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 zero-dimensional Hausdorff local compactifications.