Vladimir Kanovei , Mikhail G. Katz and Tahl Nowik

Metric completions, the Heine-Borel property, and approachability

De Gruyter | Published online: March 26, 2020

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.

MSC 2010: 53A99; 26E35; 03H05

1 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 xM is approachable if for each ε ∈ ℝ+ there is some (standard) xεM such that d (x, xε) < 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 𝔥ℝ ⊆ ℝ the subring consisting of finite hyperreals. The ring 𝔥ℝ is the domain of the standard part function st: 𝔥ℝ → ℝ. Here st(x) for x𝔥ℝ is the real number corresponding to the Dedekind cut on ℝ defined by x, via the embedding ℝ ↪ 𝔥ℝ.

Let ℚ ⊆ ℝ be the subfield consisting of hyperrational numbers. Let Fℚ be the ring of finite hyperrationals, so that F = ℚ ∩ 𝔥ℝ. Let IF be the ideal of hyperrational infinitesimals. If xℚ then its halo is the (co)set hal(x) = x + Iℚ. The following result is well known; see e.g., [2].

Theorem 1.1

The ideal IF is maximal, and the quotient field ℚ̂ = F/I is naturally isomorphic to ℝ, so that we have a short exact sequence 0 → IF → ℝ → 0.

Proof

A typical element of ℚ̂ is a halo, namely hal(x) ⊆ ℚ, where each xF can be viewed as an element of the larger ring 𝔥ℝ ⊆ ℝ. Then the map

ϕ ( hal ( x ) ) = s t ( x )

is the required map ϕ: ℚ̂ → ℝ. To show surjectivity of ϕ, note that over ℝ we have

( ϵ R + ) ( y R ) ( q Q ) [ | y q | < ϵ ] . (1.1)

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

( ϵ R + ) ( y R ) ( q Q ) [ | y q | < ϵ ] . (1.2)

Now choose an infinitesimal ϵ > 0. Then formula (1.2) implies that for each real number y there is a hyperrational q with yq, where ≈ is the relation of infinite proximity (i.e., yq 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].

2 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 xM 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 FM 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 . 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.

Definition 2.1

We let be the halo of xM.

In general the ihull of a metric space M consists of halos , where xF, with distance d on defined to be

Note that that M may be viewed as a subset of . Hence it is meaningful to speak of the closure of M in . 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 , and coincides with the closure cl(M) of M in ; see [1, Chapter 18] for a detailed discussion.

3 Universal cover of plane with a puncture

The ihull 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 dx2 + dy2 in the (x, y)-plane, which can be written in polar coordinates (r, θ) as dr2 + r22 where θ is the usual polar angle in ℝ/2πℤ.

Definition 3.1

Let M be the metric universal cover of ℝ2 ∖ {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

d r 2 + r 2 d ζ 2 . (3.1)

Formula (3.1) provides a description of the metric universal cover of the flat metric on ℝ2 ∖ {0}, for which the covering map M → ℝ2 ∖ {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.

Theorem 3.2

Points of the form for appreciable r and infinite ζ are in the ihull but are not approachable from M.

Proof

The distance function d of M extends to the ihull (, d) as in (2.1). Here points of are halos in the finite part of M. Notice that in 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 d from the point (ϵ, 0); namely the standard part of the distance is 1. Indeed, the triangle inequality applied to the sequence of points (1, ζ), ( 1 ζ 2 , ζ ) , ( 1 ζ 2 , 0 ) , (ϵ, 0) yields the bound

d ( ( 1 , ζ ) , ( ϵ , 0 ) ) ( 1 1 ζ 2 ) + 1 ζ 2 ζ + | 1 ζ 2 ϵ | 1.

Therefore

by (2.1). Hence .

On the other hand, let us show that the point is not approachable from M. Consider the rectangle K defined by the image in of

[ 1 2 , 2 ] × [ ζ 1 , ζ + 1 ] M .

The metric d of restricted to the rectangle K dominates the product metric dr2 + 1 4 2 by (3.1). The boundary of K separates the interior of K from the complement of K. Thus, to reach the finite part one must first traverse the boundary. Therefore K includes the metric ball of radius 1 2 centered at . This ball contains no standard points. Hence the point is not approachable.□

Note that what is responsible for the inapprochability is the fact that the closure 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 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, the Heine-Borel property is violated. In Section 4 we show that such a result holds more generally.

4 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 xM). 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].

Proposition 4.1

Let M be a metric space. Then the following two properties are equivalent:

  1. Every approachable point in M is nearstandard;

  2. M is complete.

Definition 4.2

A metric space M is Heine-Borel (HB) if every closed and bounded subset of M is compact.

We fix a point pM. Let n ∈ ℕ. Let Bn = {xM : d(x, p) ≤ n}. Clearly M is HB if and only if the sets Bn are all compact. Let M be the completion of M. Let Bn = {xM : d(x, p) ≤ n}, for the same fixed pM. By transfer, we have Bn = {xM : d(x, p) ≤ n}, and similarly Bn = {xM : 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).

Theorem 4.3

Let M be a metric space. The following three properties are equivalent:

  1. Every finite point in M is approachable;

  2. The completion M is Heine-Borel;

  3. M = (the metric completion is already all of the ihull).

Proof

Assume M is HB. Let aM be finite. Then we have aBnBn for some n ∈ ℕ. Since M is assumed to be HB, the ball Bn is compact. Hence there is a point xBn with xa. Now let ϵ > 0 be standard. Since M is dense in M there is a point yM 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 aBn and fix a standard ϵ > 0. Since M is dense in M, the following holds for our fixed n and ϵ:

( x B ¯ n ) ( y B n + 1 ) [ d ( x , y ) < ϵ ] .

By transfer we obtain

( x B ¯ n ) ( y B n + 1 ) [ d ( x , y ) < ϵ ] . (4.1)

Applying (4.1) with x = a, we obtain a point bBn+1 with d(a, b) < ϵ. Every finite point in M is approachable by assumption. Therefore there is a point xMM 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 Bn is compact. Let aBn. We need to find a point xBn with xa. We have shown above that a is approachable. By Proposition 4.1, a is nearstandard, i.e., there is a point xM with xa. Since d(a, p) ≤ n, and x, p are both standard, we also have d(x, p) ≤ n, i.e., xBn.□

Combining Proposition 4.1 and Theorem 4.3, we obtain the following corollary, which also appears in Goldbring [20, Proposition 9.23].

Corollary 4.4

Let M be a metric space. The following two properties are equivalent:

  1. Every finite point in M is nearstandard;

  2. M is Heine-Borel.

Acknowledgements

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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Received: 2019-10-16
Accepted: 2020-02-18
Published Online: 2020-03-26

© 2020 Vladimir Kanovei et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.