nLab
prometric space

Prometric spaces

Definitions

Let XX be an abstract set. For purposes of this definition, let a distance function on XX be a nonnegative-extended-real-valued binary function on XX; that is, a function d:X×X[0,]d\colon X \times X \to [0,\infty].

(For most purposes, we may assume that these distance functions are pointwise-bounded: taking only finite values. On the other hand, for full generality in constructive mathematics, we must allow the distance functions to take nonnegative extended upper real values, although again we may assume them to be pointwise-bounded and pointwise-located for many purposes.)

Let GG be a collection of such distance functions.

Elementary axioms

Consider the following potential properties of GG:

  1. Reflexivity: For every dGd \in G and xXx \in X, we have d(x,x)=0d(x,x) = 0.

  2. Transitivity (a version of the triangle identity): For any dGd \in G there exists an eGe \in G with

    d(x,z)e(x,y)+e(y,z) d(x,z) \leq e(x,y) + e(y,z)

    for all x,y,zXx,y,z \in X.

  3. Symmetry: For every dGd \in G, there is an eGe \in G with d(x,y)e(y,x)d(x,y) \leq e(y,x) for all x,yXx,y \in X. (In light of Isotony below, we may require d(x,y)=e(y,x)d(x,y) = e(y,x).)

  4. Nontriviality: There exists a dGd \in G. (In light of Isotony below, we may require d(x,y)=0d(x,y) = 0 for all x,yXx,y \in X.)

  5. Filtration: For all d,eGd,e \in G, there is an fGf \in G with d(x,y)f(x,y)d(x,y) \leq f(x,y) and e(x,y)f(x,y)e(x,y) \leq f(x,y) for all x,yXx,y \in X. (In light of Isotony below, we may require f(x,y)=max(d(x,y),e(x,y))f(x,y) = \max(d(x,y), e(x,y)) for all x,yXx,y \in X.)

  6. Isotony: If dGd \in G and ee is a distance function with e(x,y)d(x,y)e(x,y) \leq d(x,y) for all x,yXx,y \in X, then eGe \in G.

Sophisticated axioms

With the aid of abstract algebra? (but nothing too fancy), we may view these same 6 axioms in another light, as follows.

First, the function set [0,] X×X[0,\infty]^{X \times X} of all distance functions d:X×X[0,]d\colon X \times X \to [0,\infty] on XX is a lattice **-monoid under these operations:

  • The order \leq is given pointwise:

    de(x,y:X),d(x,y)e(x,y). d \leq e \;\iff\; \forall(x,y\colon X),\; d(x,y) \leq e(x,y) .
  • The corresponding join operation \vee is also pointwise:

    (de)(x,y)max(d(x,y),e(x,y)). (d \vee e)(x,y) \coloneqq \max(d(x,y), e(x,y)) .
  • The corresponding bottom element is the zero function 00:

    0(x,y)0. 0(x,y) \coloneqq 0 .
  • The monoid operation \circ is defined so:

    (de)(x,z)inf yX(d(x,y)+e(y,z)). (d \circ e)(x,z) \coloneqq \inf_{y \in X} (d(x,y) + e(y,z)) .
  • The corresponding identity element is the infinite Kronecker delta δ\delta:

    δ(x,y){0 ifx=y, ifxy. \delta(x,y) \coloneqq \begin{cases} 0 & if\; x = y,\\ \infty & if\; x \ne y. \end{cases}

    (This works in constructive mathematics because we are using extended upper reals; δ(x,y)\delta(x,y) is the infimum of the set {t:|t=0x=y}\{ t\colon \mathbb{R} \;|\; t = 0 \;\wedge\; x = y \}.)

  • The involution dd opd \mapsto d^{\op} is defined so:

    d op(x,y)d(y,x). d^{\op}(x,y) \coloneqq d(y,x) .

Then the same 6 axioms may be expressed as follows:

  1. Reflexivity: For every dGd \in G, we have dδd \leq \delta.

  2. Transitivity: For every dGd \in G, there exists eGe \in G such that deed \leq e \circ e.

  3. Symmetry: For every dGd \in G, there exists eGe \in G such that de opd \leq e^{\op}. (In light of Isotony, we may require d=e opd = e^op; in other words, d opGd^op \in G.)

  4. Nontriviality: There exists a dGd \in G. (In light of Isotony, we may require d=0d = 0.)

  5. Filtration: For all d,eGd,e \in G, there is an fGf \in G with dfd \leq f and efe \leq f. (In light of Isotony, we may require f=def = d \vee e.)

  6. Isotony: If dGd \in G and ee is a distance function with ede \leq d, then eGe \in G.

It is possible to generate these even more systematically by specifying a proarrow equipment and considering the (possibly symmetric) pro-monads in it; see the Categorial interpretation below.

Unbiased versions

Reflexivity and Transitivity are a binary–nullary pair whose unbiased combination is as follows:

  • Elementary version: For every natural number n=0,1,2,n = 0,1,2,\ldots and every dGd \in G, there exists eGe \in G with

    d(x 0,x n) i<ne(x i,x i+) d(x_0,x_n) \leq \sum_{i \lt n} e(x_i,x_{i+})

    for each list x 0,,x nXx_0,\ldots,x_n \in X.

  • Sophisticated version: For every natural number n=0,1,2,n = 0,1,2,\ldots and every dGd \in G, there exists eGe \in G such that de nd \leq e^{\circ n}.

We can combine these with Symmetry by generalizing nn from a natural number to a list ϵ\epsilon of bits and replacing e(x i,x i+)e(x_i,x_{i+}) with e(x i+,x i)e(x_{i+},x_i) (that is, replacing ee with e ope^{\op} in the iith position) when ϵ i=1\epsilon_i = 1.

Similarly, Nontriviality and Filtration are a binary–nullary pair whose unbiased version states (in light of Isotony) closure under finitary joins; but this is properly discussed at ideal.

Structures and spaces

A collection GG of distance functions that satisfies all of (1–6) is a prometric; if we drop (3), then we still have a quasi-prometric. If we are working with quasi-prometrics generally, then one that happens to satisfy (3) is called symmetric; in other words, a prometric is precisely a symmetric quasi-prometric.

Finally, a prometric space is a set equipped with a prometric, and likewise a quasi-prometric space is a set equipped with a quasi-prometric.

Generating (quasi)-prometrics

We sometimes wish to consider collections of distance functions that generate (quasi)-prometrics. In the following table, a collection GG satisfying the conditions listed on the left has the name on the right:

ConditionsName
1,2pre-quasi-prometric
1,2,3pre-prometric
      4,5ideal base
1,2,  4,5quasi-prometric base
1,2,3,4,5prometric base
      4,5,6ideal
1,2,  4,5,6quasi-prometric
1,2,3,4,5,6prometric

Here we always use the original (either elementary or sophisticated) formulation of (3–5); one might consider what it means if these satisfy the stronger versions rewritten in light of Isotony, but it's getting a bit far along into centipede mathematics to actually give these things names.

A (quasi)-prometric base is precisely an ideal base whose generated ideal is a (quasi)-prometric. We may also speak of a (quasi)-prometric subbase as an ideal subbase (that is, an arbitrary collection) whose generated ideal base is a (quasi)-prometric base, or equivalently whose generated ideal is a (quasi)-prometric. A pre-(quasi)-prometric is always a (quasi)-prometric subbase, but not conversely; but thinking too hard about subbases risks more centipedes.

A pre-quasi-prometric is symmetric if its generated quasi-prometric is symmetric (hence a prometric); a quasi-prometric base is symmetric iff it is a prometric base (and the analogous result holds for subbases), but a symmetric pre-quasi-prometric need not be a pre-prometric. Some authors may require a strong version of Symmetry in which the distance functions dd are all individually required to be symmetric; that is, d(x,y)=d(y,x)d(x,y) = d(y,x) (or simply d=d opd = d^{\op}). Every prometric has a base with this property; in particular, if GG is a prometric, then

{dG|d=d op} \{ d \in G \;|\; d = d^{\op} \}

is a prometric base that generates GG, and it is this base that some authors may refer to as the prometric itself. (I write ‘some authors may’, but there are few authors on this subject; still, it's the sort of thing that somebody might do.)

A pre-quasi-prometric GG is pointwise-bounded if every dGd \in G is pointwise-bounded; that is, d(x,y)<d(x,y) \lt \infty for every x,yXx,y \in X (or simply d<d \lt \infty). If GG is a (quasi)-prometric, then

G b{dG|d<} G_b \coloneqq \{ d \in G \;|\; d \lt \infty \}

is a (quasi)-prometric that is very similar to GG (in particular, they are uniformly equivalent?), and some authors may prefer to work with G bG_b. Indeed, when pointwise boundedness is not required, some authors may call the structure extended, as an instance of the red herring principle. However, requiring pointwise boundedness interferes with the more sophisticated approaches to (quasi)-prometrics; in particular, δ\delta is not pointwise-bounded. (It is possible that prometric spaces would work better with a closure condition that would make G bG_b generate GG.)

Most definitions are no more complicated when phrased in terms of (quasi)-prometric bases or even pre-(quasi)-prometrics, and some constructions give one of these more naturally than the generated (quasi)-prometric. When working in predicative mathematics, it is preferable to work exclusively with (quasi)-prometric bases, as the generated (quasi)-prometric will typically be a proper class. However, it is ultimately the generated (quasi)-prometric (even if referred to only obliquely) that matters.

Any (quasi)-gauge is a (quasi)-prometric base; similarly, given any (quasi)-pseudometric dd, the singleton {d}\{d\} is a (quasi)-prometric base (and its generated (quasi)-gauge in turn generates the same (quasi)-prometric). One might call a (quasi)-prometric (space) simple if it is generated in this way by a (quasi)-pseudometric; this term is used analogously in the theory of syntopogenous spaces, but the term (quasi)-pseudometrizable is more likely to be understood. (See Subcategories below.)

Morphisms

A short map between (quasi)-prometric spaces XX and YY is a function f:XYf:X\to Y such that for every dG Yd\in G_Y, we have d(f×f)G Xd\circ (f\times f) \in G_X. We write ProMetProMet for the category of prometric spaces and short maps, and similarly QProMetQProMet for the category of quasi-prometric spaces and short maps.

If the (quasi)-prometrics of XX and YY are presented by bases, then this is equivalent to saying that for any basic distance function dd on YY, there is a basic ee on XX such that d(f(x),f(x))e(x,x)d(f(x),f(x'))\le e(x,x') for all x,xXx,x'\in X. Thus, for (quasi)-pseudometric spaces and (quasi)-gauge spaces considered as (quasi)-prometric spaces, this reduces to the usual notion of short map (i.e., distance-decreasing map). Hence the category GauGau of gauge spaces and short maps is included as a full subcategory of ProMetProMet, and similarly with QGauQGau in QProMetQProMet.

Subcategories

Since GauGau includes the categories of metric spaces and uniform spaces (disjointly), so does ProMetProMet. Likewise, since QGauQGau includes the category of topological spaces (disjointly from metric and uniform spaces), so does QProMetQProMet.

There is also another embedding of UnifUnif into ProMetProMet, however, which is notably simpler than its embedding into GauGau (both easier to state and not using dependent choice). Given a uniform space XX, we define for each entourage UX×XU\subseteq X\times X a distance function

d U(x,y)={0 (x,y)U (x,y)U. d_U(x,y) = \begin{cases} 0 & (x,y)\in U\\ \infty & (x,y)\notin U. \end{cases}

(Again, this may be defined constructively as an infimum.) The collection of such d Ud_U is a base for a prometric on XX. The short maps between such prometric spaces are precisely the uniformly continuous ones, so this defines another embedding of UnifUnif into ProMetProMet. The full image of this embedding consists precisely of those prometric spaces generated by a base of {0,1}\{0,1\}-valued functions. Note that replacing \infty by any positive real number also defines an embedding of UnifUnif into ProMetProMet, but a yet different one.

Conversely, every prometric induces a uniformity, where the entourages are the sets

U d,ϵ={(x,y)|d(x,y)<ϵ}. U_{d,\epsilon} = \{ (x,y) \;|\; d(x,y) \lt \epsilon \} .

In this way every short map induces a uniformly continuous map as well. This operation is compatible with the above inclusions of UnifUnif, as well as with the inclusion via GauGau.

Categorical interpretation

As observed by Lawvere, an extended quasi-pseudo-metric space is a category enriched over the monoidal category ([0,],,+,0)([0,\infty],\geq,+,0). In other words, it is a monoid (or monad) in the bicategory [0,]Mat[0,\infty] Mat of matrices with values in this monoidal category. Analogously, an extended quasi-prometric space is a monad in the bicategory Pro[0,]MatPro [0,\infty] Mat whose hom-categories are the categories of pro-objects in the hom-categories of [0,]Mat[0,\infty] Mat.

Note that if Rel={0,1}MatRel = \{0,1\} Mat denotes the bicategory of relations in SetSet, then a monad in RelRel is a preorder, while a monad in ProRelPro Rel is a quasi-uniform space.

In all these cases, in order to recover the correct notion of morphism abstractly, we must consider monads in a double category or equipment rather than merely a bicategory.

Generalized uniform structures

proarrowmonadpro-monadsymmetric versions
binary relationpreorderquasiuniformitysymmetric relationequivalence relationuniformity
binary function to [0,)[0,\infty)quasipseudometricquasiprometricsymmetric binary functionpseudometricprometric
topogenyquasiproximitysyntopogenysymmetric topogenyproximitysymmetric syntopogeny

References

  • Maria Manuel Clementino, Dirk Hofmann, and Walter Tholen. “One Setting for All: Metric, Topology, Uniformity, Approach Structure.” (pdf)

Revised on August 29, 2014 19:44:15 by Toby Bartels (98.16.178.155)