Abstract
We introduce the notion of Hausdorff extension of an arbitrary set X and we study the connections with the Stone-Cech compactification \beta X of the discrete space X. We characterize those Hausdorff extensions that satisfy the “transfer principle” of nonstandard analysis, and we investigate the consistency strength of their existence.
Highlights
In this paper we prove that a topological model of the hypernatural numbers is a nonstandard model of the natural numbers
Topological nonstandard extensions are studied in full generality in [9]
It turns out that the existence of a Hausdorff model of the hypernatural numbers is equivalent to the existence of an ultrafilter satisfying a property, labelled (C) in [8], which has been rarely considered in the literature
Summary
In this paper we prove that a topological model of the hypernatural numbers is a nonstandard model of the natural numbers. The similarity with the continuous extensions of functions in various compactifications of topological spaces suggests the following definition of Hausdorff extension of an arbitrary set X. Lemma 1.1 The following properties hold in any Hausdorff extension ∗X of X, for all functions f, g : X → X: 1.
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