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

Generalizing a theorem of Ph. Dwinger [7], we describe the 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 two given Hausdorff zerodimensional 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 [1] about the maximal zero-dimensional compactification. Extending a recent theorem of G. Bezhanishvili [2], we describe the local proximities corresponding to the zero-dimensional Hausdorff local compactifications. MSC: primary 54C20, 54D35; secondary 54C10; 54D45.