Compactifications of metric spaces
Let \((X, d)\) be a metric space. Then a filter \({\mathcal G}\) on \(X\) is called a near ultrafilter on \((X, d)\) if it is maximal with respect to the near finite intersection property (i.e., for every finite \({\mathcal F}\subseteq{\mathcal G}\) and every \(\varepsilon>0\), \(\bigcap\{B(Y,\varepsilon):Y\in{\mathcal F}\}\neq\emptyset\), where \(B(Y,\varepsilon)=\{x\in X:d(x, Y)<\varepsilon\}\)). The authors investigate the space \(\widetilde{X}\) of all near ultrafilters on \((X,d)\) with the natural topology. They prove that the space \(\widetilde{X}\) is a Hausdorff compactification of \((X,d)\) with the following two remarkable properties: (1) If \(X\) and \(Y\) are metric spaces and \(f:X\to Y\) is a uniformly continuous map, then there exists a continuous extension \(\widetilde{f}:\widetilde{X}\to \widetilde{Y}\) of \(f\) such that \(\widetilde{f}\circ e_X=e_Y\circ f\), where \(e_X:X\to\widetilde{X}\) and \(e_Y:Y\to\widetilde{Y}\) are natural embeddings. (2) A bounded real-valued continuous function \(f:X\to{\mathbb R}\) has a continuous extension \(\widetilde{f}:\widetilde{X}\to{\mathbb R}\) if and only if \(f\) is uniformly continuous. The second property implies that the algebra \(C(\widetilde{X})\) of all real-valued continuous functions on \(\widetilde{X}\) is isomorphic to that of all real-valued uniformly continuous functions on \(X\). It is also shown that if \((S, d)\) is a metric space such that \(S\) is a semigroup and \(d\) is an invariant metric, then \(\widetilde{S}\) is a semigroup compactification of \((S,d)\).
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