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Abstract
Abstract We show that the metric universal cover of a plane with a puncture yields an example of a nonstandard hull properly containing the metric completion of a metric space. As mentioned by Do Carmo, a nonextendible Riemannian manifold can be noncomplete, but in the broader category of metric spaces it becomes extendible. We give a short proof of a characterisation of the Heine-Borel property of the metric completion of a metric space M in terms of the absence of inapproachable finite points in ∗ M.
Highlights
We show that the metric universal cover of a plane with a puncture yields an example of a nonstandard hull properly containing the metric completion of a metric space
We give a short proof of a characterisation of the Heine-Borel property of the metric completion of a metric space M in terms of the absence of inapproachable nite points in *M
A p-adic power series example of the phenomenon of inapproachability in a nonstandard hull of a metric space M appears in Goldblatt [1, p. 252]
Summary
A p-adic power series example of the phenomenon of inapproachability in a nonstandard hull of a metric space M appears in Goldblatt [1, p. 252]. A p-adic power series example of the phenomenon of inapproachability in a nonstandard hull of a metric space M appears in Goldblatt [1, p. A nonstandard hull of a metric space M can in general contain points that need to be discarded (namely, the inapproachable ones) in order to form the metric completion of M. Let *R be a hyperreal eld extending R. Let *Q ⊆ *R be the sub eld consisting of hyperrational numbers. Let F ⊆ *Q be the ring of nite hyperrationals, so that F = *Q ∩ hR. Let I ⊆ F be the ideal of hyperrational in nitesimals. A typical element of Q is a halo, namely hal(x) ⊆ *Q, where each x ∈ F can be viewed as an element of the larger ring hR ⊆ *R.
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