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.
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
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].
The ideal I ⊆ F is maximal, and the quotient field ℚ̂ = F/I is naturally isomorphic to ℝ, so that we have a short exact sequence 0 → I → F → ℝ → 0.
A typical element of ℚ̂ is a halo, namely hal(x) ⊆ ^{∗}ℚ, where each x ∈ F can be viewed as an element of the larger ring ^{𝔥}ℝ ⊆ ^{∗}ℝ. Then the map
is the required map ϕ: ℚ̂ → 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 ϵ > 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].
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
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
is called the ihull of M and denoted M̂. 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.
We let be the halo of x ∈ ^{∗}M.
In general the ihull M̂ of a metric space M consists of halos , where x ∈ F, with distance d on M̂ defined to be
Note that that M may be viewed as a subset of M̂. Hence it is meaningful to speak of the closure of M in M̂. We will denote such closure cl(M) to distinguish it from the abstract notion of the metric completion M of M (see Section 4). The closure cl(M) is indeed the completion in the sense that it is complete and M is dense in it. The metric completion M of M is the approachable part of M̂, and coincides with the closure cl(M) of M in M̂; see [1, Chapter 18] for a detailed discussion.
The ihull M̂ 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.
Let M be the metric universal cover of ℝ^{2} ∖ {0} (the plane minus the origin), coordinatized by (r, ζ) where r > 0 and ζ is an arbitrary real number.
In formulas, M can be given by the coordinate chart r > 0, ζ ∈ ℝ, equipped with the metric
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.
Points of the form for appreciable r and infinite ζ are in the ihull M̂ but are not approachable from M.
The distance function d of M extends to the ihull (M̂, d) as in (2.1). Here points of M̂ are halos in the finite part of ^{∗}M. Notice that in M̂ the origin has been “restored" and can be represented in coordinates (r, ζ) by a point (ϵ, 0) in ^{∗}M where ϵ > 0 is infinitesimal.
Consider a point (1, ζ) ∈ ^{∗}M where ζ is infinite. Let us show that the point (1, ζ) is at a finite distance
Therefore
by (2.1). Hence ∈ M̂.
On the other hand, let us show that the point ∈ M̂ is not approachable from M. Consider the rectangle K defined by the image in M̂ of
The metric d of M̂ restricted to the rectangle K dominates the product metric dr^{2} +
Note that what is responsible for the inapprochability is the fact that the closure M ⊆ M̂ 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 is a line.
Do Carmo [14, 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 ∈ M ⊆ M̂, the Heine-Borel property is violated. In Section 4 we show that such a result holds more generally.
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 [18, Theorem 3.14.1, p. 78] and Hurd-Loeb [19, Proposition 3.14]; cf. Davis [2, Theorems 5.19 and 5.20, p. 93].
Let M be a metric space. Then the following two properties are equivalent:
Every approachable point in ^{∗}M is nearstandard;
M is complete.
A metric space M is Heine-Borel (HB) if every closed and bounded subset of M is compact.
We fix a point p ∈ M. Let n ∈ ℕ. Let B_{n} = {x ∈ M : d(x, p) ≤ n}. Clearly M is HB if and only if the sets B_{n} are all compact. Let M be the completion of M. Let B_{n} = {x ∈ M : d(x, p) ≤ n}, for the same fixed p ∈ M. By transfer, we have ^{∗}B_{n} = {x ∈ ^{∗}M : d(x, p) ≤ n}, and similarly ^{∗}B_{n} = {x ∈ ^{∗}M : d(x, p) ≤ n}.
Clearly, an HB metric space is complete (given a Cauchy sequence, find a convergent subsequence in the closure of its set of points).
Let M be a metric space. The following three properties are equivalent:
Every finite point in ^{∗}M is approachable;
The completion M is Heine-Borel;
M = M̂ (the metric completion is already all of the ihull).
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 ϵ > 0 be standard. Since M is dense in M there is a point y ∈ M such that d(y, x) < ϵ, and therefore d(y, a) < ϵ.
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 ϵ > 0. Since M is dense in M, the following holds for our fixed n and ϵ:
By transfer we obtain
Applying (4.1) with x = a, we obtain a point b ∈ ^{∗}B_{n+1} with d(a, b) < ϵ. Every finite point in ^{∗}M is approachable by assumption. Therefore there is a point x ∈ M ⊆ M with d(x, b) < ϵ. Thus d(x, a) < 2ϵ, 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].
Let M be a metric space. The following two properties are equivalent:
Every finite point in ^{∗}M is nearstandard;
M is Heine-Borel.
We are grateful to the referees for helpful comments on an earlier version of the manuscript. V. Kanovei was partially supported by RFBR grant no 18-29-13037.
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