  Generalizing a theorem of Ph. Dwinger, 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 given in advance Hausdorff zero-dimensional local
compactifications of these spaces; we regard the following kinds of extensions:
continuous, open, quasi-open, skeletal, perfect, injective, surjective. In this
way we generalize some classical results of B. Banaschewski about the maximal
zero-dimensional Hausdorff compactification. Extending a recent theorem of G.
Bezhanishvili, we describe the local proximities corresponding to the
zero-dimensional Hausdorff local compactifications.
