Metric completions, the Heine-Borel property, and approachability

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.


Introduction
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]. Recall that a point x ∈ * M is approachable if for each ε ∈ R + there is some (standard) x ε ∈ M such that * d(x, x ε ) < ε (op. cit., p. 236). Otherwise x is called inapproachable.
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. We provide a more geometric example of such a phenomenon stemming from differential geometry. The example is the metric universal cover of a plane with one puncture; see Definition 3.1.
Let * R be a hyperreal field extending R. Denote by h R ⊆ * R the subring consisting of finite hyperreals. The ring h R is the domain of the standard part function st : h R → R. Here st(x) for x ∈ h R is the real number corresponding to the Dedekind cut on R defined by x, via the embedding R ֒→ h R.
Let * Q ⊆ * R be the subfield consisting of hyperrational numbers. Let F ⊆ * Q be the ring of finite hyperrationals, so that F = * Q ∩ h R. Let I ⊆ F be the ideal of hyperrational infinitesimals. If x ∈ * Q then its halo is the (co)set hal(x) = x + I ⊆ * Q. The following result is well known; see e.g., [2]. Theorem 1.1. The ideal I ⊆ F is maximal, and the quotient field Q = F/I is naturally isomorphic to R, so that we have a short exact se- Proof. 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 h R ⊆ * R.
is the required map φ : Q → R. To show surjectivity of φ, note that over R we have Recall that Robinson's transfer principle (see [3]) asserts that every first-order formula, e.g., (1.1), has a hyperreal counterpart obtained by starring the ranges of the bound variables. We apply the upward transfer principle to (1.1) to obtain Now choose an infinitesimal ǫ > 0. Then formula (1.2) implies that for each real number y there is a hyperrational q with y ≈ q, where ≈ is the relation of infinite proximity (i.e., y − q is infinitesimal). Therefore we obtain φ(hal(q)) = y, as required.
A framework for differential geometry via infinitesimal displacements was developed in [4]. An application to small oscillations of the pendulum appeared in [5]. The reference [1] is a good introductory exposition of Robinson's techniques, where the reader can find the definitions and properties of the notions exploited in this text. Additional material on Robinson's framework can be found in [6]. The historical significance of Robinson's framework for infinitesimal analysis in relation to the work of Fermat, Gregory, Leibniz, Euler, and Cauchy has been analyzed respectively in [7], [8], [9], [10], [11], and elsewhere. The approach is not without its critics; see e.g., [12].

Ihull construction
In Section 1 we described a construction of R starting from * Q. More generally, one has the following construction. In the literature this construction is often referred to as the nonstandard hull construction, which we will refer to as the ihull construction ("i" for infinitesimal ) for short. The general construction takes place in the context of an arbitrary metric space M.
Given a metric space (M, d), we consider its natural extension * M. The distance function d extends to a hyperreal-valued function * d on * M as usual. The halo of x ∈ * M is defined to be the set of points in * M at infinitesimal distance from x.
Let ≈ be the relation of infinite proximity in * M . Denote by F ⊆ * M the set of points of * M at finite distance from any point of M (i.e., the galaxy of any element in M). The quotient F/ ≈ is called the ihull of M and denoted M . In this terminology, Theorem 1.1 asserts that the ihull of Q is naturally isomorphic to R. Thus, ihulls provide a natural way of obtaining completions; see Morgan ([13], 2016) for a general framework for completions. We will exploit the following notation for halos.

Universal cover of plane with a puncture
The ihull M may in general be larger than the metric completion M of M. An example of such a phenomenon was given in [1, p. 252] in terms of p-adic series. We provide a more geometric example of such a phenomenon stemming from differential geometry.
We start with the standard flat metric dx 2 + dy 2 in the (x, y)-plane, which can be written in polar coordinates (r, θ) as dr 2 + r 2 dθ 2 where θ is the usual polar angle in R/2πZ. In formulas, M can be given by the coordinate chart r > 0, ζ ∈ R, equipped with the metric dr 2 + r 2 dζ 2 . (3.1) Formula (3.1) provides a description of the metric universal cover of the flat metric on R 2 \ {0}, for which the covering map M → R 2 \ {0} sending (r, ζ) → (r, θ) induces a local Riemannian isometry, where θ corresponds to the coset ζ + 2πZ. Recall that a number is called appreciable when it is finite but not infinitesimal. Note that what is responsible for the inapprochability is the fact that the closure M ⊆ M does not have the Heine-Borel property (and is not even locally compact). Namely, the boundary of the metric unit ball in M centered at the origin (ǫ, 0) is a line.
Do Carmo ( [14], 1992, p. 152) views the universal cover M of the plane with a puncture as an example of a Riemannian manifold that is nonextendible but not complete. Indeed, M is nonextendible in the category of Riemannian manifolds, but M is extendible in the category of metric spaces, in such a way that near the "extended" origin (ǫ, 0) ∈ M ⊆ M , the Heine-Borel property is violated. In Section 4 we show that such a result holds more generally.

An approachable criterion for the Heine-Borel property
For the sake of completeness we provide a short proof of a relation between approachability and the Heine-Borel property for metric spaces. For related results in the context of uniform spaces see Henson-Moore [15], [16] (but note that they use a different notion of "finiteness" for a point x ∈ * M ). For a study of the relation between the Heine-Borel property and local compactness, see [17].
We show that the Heine-Borel property for the completion of a metric space has a characterisation in terms of the absence of finite inapproachable points; see Theorem 4.3. The following result appears in Luxemburg [  (1) every approachable point in * M is nearstandard; (2) M is complete. Clearly, an HB metric space is complete (given a Cauchy sequence, find a convergent subsequence in the closure of its set of points). Proof. Assume M is HB. Let a ∈ * M be finite. Then we have a ∈ * B n ⊆ * B n for some n ∈ N. Since M is assumed to be HB, the ball B n is compact. Hence there is a point x ∈ B n with x ≈ a. Now let ǫ > 0 be standard. Since M is dense in M there is a point y ∈ M such that d(y, x) < ǫ, and therefore d(y, a) < ǫ.
Conversely, assume every finite point in * M is approachable. As a first step we show that every finite point in * M is approachable. Let a ∈ * B n and fix a standard ǫ > 0. Since M is dense in M , the following holds for our fixed n and ǫ: By transfer we obtain (∀x ∈ * B n )(∃y ∈ * B n+1 )[d(x, y) < ǫ]. (4.1) Applying (4.1) with x = a, we obtain a point b ∈ * B n+1 with d(a, b) < ǫ. Every finite point in * M is approachable by assumption. Therefore there is a point x ∈ M ⊆ M with d(x, b) < ǫ. Thus d(x, a) < 2ǫ, showing that every finite point in * M is approachable. We now prove that M is HB by showing that each B n is compact. Let a ∈ * B n . We need to find a point x ∈ B n with x ≈ a. We have shown above that a is approachable. By Proposition 4.1, a is nearstandard, i.e., there is a point x ∈ M with x ≈ a. Since d(a, p) ≤ n, and x, p are both standard, we also have d(x, p) ≤ n, i.e., x ∈ B n . Combining Proposition 4.1 and Theorem 4.3, we obtain the following corollary, which also appears in Goldbring [20,Proposition 9.23]. (1) Every finite point in * M is nearstandard; (2) M is Heine-Borel.