Topological and nonstandard extensions

We introduce a notion of topological extension of a given set X. The resulting class of topological spaces includes the Stone-ˇCech compactification βX of the discrete space X, as well as all nonstandard models of X in the sense of nonstandard analysis (when endowed with a “natural” topology). In this context, we give a simple characterization of nonstandard extensions in purely topological terms, and we establish connections with special classes of ultrafilters whose existence is independent of ZFC.