Completion of premetric spaces

We present a method for completing a premetric space, in the sense introduced by F. Richman in the context of constructive Mathematics without countable choice.


Introducción
F. Richman [8] addressed the problem of completing a metric space in constructive mathematics without the axiom of countable choice. In constructive mathematics the underlying logic is intuitionistic instead of classical. The axiom of countable choice (CC) says that any countable collection of non empty sets (X n ) n has a choice function, i.e., a function f : N → n X n such that f (n) ∈ X n for all n ∈ N. This weak form of choice suffices for the proof of several classical results, for instance, CC is equivalent to the Baire category theorem for certain class of metric spaces (the full Baire theorem requires the axiom of dependent choice, see [2] for a complete treatment of these issues). F. Richman [7] presented an interesting account of the arguments in favor and also against CC in the context of constructive mathematics. In a choiceless environment, he argued that the reals numbers should not be defined as Cauchy sequences [6,8]. Lubarsky has shown [3] that in constructive mathematics without choice it cannot be proved that the Cauchy reals are complete. To point out why CC is needed, we recall briefly, for the readers convenience, the usual completion of a metric space (see for instance [10]). Given a metric space X, we consider the quotient Z obtained by setting in the set of all Cauchy sequences in X, the natural equivalence relation where two Cauchy sequences are equivalent if its elements get as closed to 0 as we wish. Then Z is endowed with a metric which makes it a complete metric space containing a dense isometric copy of X. To prove that Z is indeed complete, we pick a sequence ([s] i ) i in Z, then the axiom of countable choice give us a sequence ( Without CC, sequences are not the proper objects to represent real numbers. Another natural approach is through Dedekind cuts which is only applicable to ordered structures (see [4,5] for a comparison of different approaches in the context of constructive mathematics). Richman [8] replaced the notion of a metric space (which assumes the existence of R) by a structure he called a premetric space which only needs the rational numbers and presented a method for completing a premetric space. He then used these ideas to define the real numbers in constructive mathematics without CC. We will present a method for completing premetric spaces similar to Richman's but simpler. Now we recall the basic definitions and the proposed completion introduced by Richman. Let X be a nonempty set. A binary relation d between X × X and the nonnegative rationals is a premetric on X, and we write d(x, y) ≤ q instead of ((x, y), q) ∈ d, if it satisfies the following conditions for all x, y, z ∈ X and all nonnegative rational numbers p, q: (1) d(x, y) ≤ 0 if and only if x = y.
(4) d(x, y) ≤ p if and only if d(x, y) ≤ q for all q > p (upper continuity).
A set with a premetric is called a premetric space, and we use the notation (X, d) to specify the set and its premetric. Also, we write d(x, y) q instead of ((x, y), q) / ∈ d. When a relation d satisfies 2, 3, 4 above and also d(x, x) ≤ 0 for all x ∈ X, then d is called a pseudo-premetric. When (X, d) is a pseudopremetric space, then the relation x ∼ y if d(x, y) ≤ 0 is an equivalence relation. Thus we can define the corresponding quotient and get a premetric space where the induced premetric is given by is an equivalence relation, and that the induced premetric is indeed a premetric depends only on the first three defining conditions of a pseudo-premetric.
In general, common notions defined for metric spaces can be copied on premetric spaces. If D ⊆ X and X is premetric space, then D is said to be dense in X when given any ε > 0 and x ∈ X, there exists y ∈ D such that d(y, x) ≤ ε. A map f : X → Y between two premetric spaces (X, d X ) and (Y, d Y ) is an isometric embedding if for all x, x ∈ X and q a non negative rational, d X (x, x ) ≤ q if, and only if, d Y (f (x), f (x )) ≤ q. When f is in addition onto, then it is called an isometry. The metric notion of diameter of a set is also defined on a premetric space as a binary relation, namely, we write diam A ≤ q for A ⊆ X and a nonnegative rational number q, if for all x, y ∈ A, d(x, y) ≤ q.
Richman introduced the following notions. A family {S q : q ∈ Q + } of subsets of X is regular if d(x, y) ≤ p + q for all x ∈ S p and y ∈ S q . Two regular families S = {S q : q ∈ Q + } and T = {t q : q ∈ Q + } are equivalent if d(x, y) ≤ p + q for all x ∈ S p and y ∈ T q . Let X R be the quotient of all regular families under that equivalence relation. The natural identification On his review of Richman's paper [8], A. Setzer [9] remarked that there was not a formal verification that ( X R , d R ) was indeed a R-complete premetric space. Motivated in part by Setzer's remark, we define a premetric space X using, instead of regular families, some collections of subsets of X similar to Cauchy filters but simpler. We introduce a notion of completeness and show that X is a completion of X. Our method is somewhat similar to that of completing a uniform space. Finally, we show that ( X R , d R ) is isometric to ours and it is R-complete, thus filling a gap left in Richman's paper.

Completeness
In order to simplify the treatment of completeness developed by Richman, we use, instead of regular families, a notion similar to that of a Cauchy filter but much weaker. Similar ideas about the completion of uniform spaces can be found in §3, Chapter II of [1].
Definition 2.1. Let X be a premetric space and F ⊆ P(X). We say that F is a Cauchy family on X if it satisfies the following: ii) for all ε > 0 there exists S ∈ F such that diam S ≤ ε.
A trivial example of Cauchy family is {{x}}, where x is any point of a premetric space. A Cauchy filter is a just a Cauchy family which is also a filter. Notice that a Cauchy family does not need to be even a filter base, as it could fail to have the finite intersection property. For instance, consider Q as a premetric space with its natural premetric given by the relation |x − y| ≤ q.
Then F is a Cauchy family that does not satisfy the finite intersection property as There is no problem in using Cauchy filters instead of regular families, but the notion of a Cauchy family is sufficient for our purposes. Now we introduce a pseudo-premetric on the collection of Cauchy families.
Definition 2.2. Let F and F be Cauchy families on X, we say that d(F, F ) ≤ q, if for all ε > 0 there exist S ∈ F and T ∈ F such that: The intuition behind the premetric on the collection of Cauchy families is as follows. Considering each Cauchy family F as a point, then the elements of F are "good approximations" of F . Thus smaller the diameter of an element S ∈ F , the better an approximation it is. Thus, roughly speaking, two Cauchy families F , F are at distance at most q, when there are approximations S and T of F and F , respectively, such that S and T are at distance at most q.
Here we abuse the notation a bit, we use d for both the premetric on X and the ternary relation just defined above. Proof. Let F and F be two Cauchy families and q a nonnegative rational number. Applying directly conditions i) and ii) from Definition 2.2 we have that To prove the triangular inequality, suppose that d(F, G) ≤ p and d(G, H) ≤ q. Then for each ε > 0 there are subsets S ∈ F , U ∈ H and T, T ∈ G such that diam S, diam U , diam T , diam T ≤ ε 2 , and such that diam (S ∪ T ) ≤ p + ε 2 and diam (T ∪ U ) ≤ q + ε 2 . By condition i) of the definition of a Cauchy family, there is t ∈ T ∩ T . Thus d(x, t) ≤ p + ε 2 and d(t, y) ≤ q + ε 2 for all x ∈ S and y ∈ U , so by the triangular inequality on X we get d(x, y) ≤ p + q + ε, and diam (S ∪ U ) ≤ p + q + ε. Also, by upper continuity on X, diam S, diam U ≤ ε, so we have shown that d(F, H) ≤ p + q.
Finally to show upper continuity, suppose first that d(F, F ) ≤ p and q > p. Then for each ε > 0 we can find S ∈ F and T ∈ F such that diam S, diam T ≤ ε, satisfying moreover that diam (S ∪ T ) ≤ p + ε. Since p + ε < q + ε the upper continuity of d on X implies that also diam (S ∪ T ) ≤ q + ε, and thus by definition, d(F, F ) ≤ q. Conversely, suppose d(F, F ) ≤ q for all q > p. Let ε > 0 and set q = p + ε 2 , there are S ∈ F and T ∈ F such that diam S, diam T ≤ ε 2 and such that diam (S ∪ T ) ≤ q + ε 2 = p + ε. By applying again upper continuity (on X) we have that diam S, diam T ≤ ε and thus d(F, F ) ≤ p. Now we introduce the premetric space of Cauchy families of a given premetric space.
Definition 2.4. Let X be a premetric space. The premetric space obtained from the quotient of the set of all Cauchy families on X, induced by its respective pseudo-premetric, is denoted by ( X, d ). Also, we use the notation x * for the equivalence class of the family {{x}}.
As usual, we denote the elements of X as [F ], the equivalence class of the Cauchy family F . Theorem 2.5. Let X be a premetric space. The function i : X → X defined as i(x) = x * is an isometric embedding with dense image.
Following Richman, we now introduce the notion of a complete premetric space, in terms of its canonical map. Definition 2.6. Let X be a premetric space. We say that X is complete, if the map i : X → X is onto.

Completion
Having at hand the formal definition of completeness, we now introduce the natural notion of a completion of a premetric space.
Definition 3.1. Let X and Y be premetric spaces. We say that Y is a completion of X, if Y is complete, and there exists an isometric embedding j : X → Y with dense image. By Theorem 2.5, the existence of a completion of a premetric space X will be immediately guaranteed once we show that ( X, d) is complete. To show that X is complete, we prove an extension theorem. We remark that the heart of the argument is this extension theorem, the rest of the proof of the completeness of X and the uniqueness of the completion will be achieved by algebraic manipulation of diagrams.
Thereby we set S c = {S c q : q ∈ Q + }. We will show that S c is a Cauchy family on B for all c ∈ C. Indeed, since h has dense image, given two nonnegative rationals µ and ν, there exists x ∈ A such that d C (h(x), c) ≤ 1 2 min{µ, ν}. Therefore f (x) ∈ S c µ ∩ S c ν and we have shown that S c µ ∩ S c ν = ∅. On the other hand, to check the second condition of a Cauchy family, it suffices to show that diam S c ε ≤ ε for all ε > 0. Let ε > 0, then for all x, y ∈ A such that d C (h(x), c) ≤ ε/2 and d C (h(y), c) ≤ ε/2, we have, by the triangular inequality, that d C (h(x), h(y)) ≤ ε. Since h and f are isometric embeddings we also have d A (x, y) ≤ ε and d B (f (x), f (y)) ≤ ε. Thus diam S c ε ≤ ε and S c is a Cauchy family. Now define g(c) = [S c ] for each c ∈ C. To prove that g is an isometric This follows immediately from the triangular inequality since for every f (x) ∈ S c ε and f (y) ∈ S c ε we have d B (f (x), f (y)) ≤ p + ε. Conversely, To see that d C (c, c ) ≤ p it suffices to show that d C (c, c ) ≤ p + ε for all ε > 0. Then, let ε > 0 and choose µ and ν such that diam (S c µ ∪ S c ν ) ≤ p + ε/3. Since the image of h is dense in A, there exist a, a ∈ A such that d C (h(a), c) ≤ 1 3 min{µ, ε} and d C (h(a ), c ) ≤ 1 3 min{ν, ε}. As d C (h(a), c) ≤ ε/3 and d C (h(a ), c ) ≤ ε/3, then h(a) ∈ S c µ and h(a ) ∈ S c ν and therefore d C (h(a), h(a )) ≤ p + ε/3. Hence by the triangular inequality, d C (c, c ) ≤ p+ε for all ε > 0, and by upper continuity we conclude d C (c, c ) ≤ p.
Finally observe that for all a ∈ A we have f (a) ∈ S h(a) ε for all ε > 0, thereby (h(a)). Therefore g • h = i • f and the diagram commutes.
Finally, we show that X is indeed a completion and it is unique up to isometry. Theorem 3.3. ( X, d) is a complete premetric space for every premetric space (X, d) and thus every premetric space admits a completion which is unique up to isometry.
Proof. We will show that X is complete. By Theorem 2.5, we have the isometric embeddings i : X → X and j : X → X each one with dense image.
Thus the map k = j • i : X → X has also dense image. Thereby we can apply the Theorem 3.2 to find a map l such that the following diagram commutes Contracting the diagram we have X X X X i k j jl By the uniqueness given by Theorem 3.2 j • l = id, and this proves that j is onto, that is, that X is complete.
To prove uniqueness, suppose Y is a complete premetric space and h : X → Y is an isometric embedding with dense image. Let j : Y → Y and i : X → X be the natural embeddings. Since Y is complete, j is a bijection. Then by Theorem 3.2 there are maps u, v such that the following diagram commutes.
By uniqueness we have v • u = id. Thus the map u is an isometry between Y and X.
As we said in the introduction, Richman proposed in [8] a way of completing a premetric space but he did not formally verified that it was indeed a complete premetric space. Now we are going to show that ( X R , d R ) is R-complete. The definition was given in the introduction.
Let X R be the quotient of all regular families under the equivalence relation defined in the introduction and d R be the premetric on X R . The natural iden- for all q ∈ Q + . Richman proved that i is an isometric embedding with dense image (see [8,Theorem 2.2]). Let X R denote the quotient of all regular families of ( X R , d R ) and i R : X R → X R the natural isometric embedding. We will show that i R is onto.
We need the following lemma proved by Richman. . Let X be a premetric space and S = {S q : q ∈ Q + } a regular family on X. Then d R (i R (x), [S]) ≤ q for any x ∈ S q and q ∈ Q + . Theorem 3.5. Let X be a premetric space, then (i) There exists an isometry ϕ : ( X R , d R ) → ( X, d).
(ii) The map i R : X R → X R is onto.
Proof. (i) As the natural map i : X → X R is an isometric embedding with dense image, then by Theorem 3.2 there exists an isometric embedding ϕ such that the next diagram commutes We prove now that ϕ is onto. Let s ∈ X, we define for each positive rational q the set Since j is an isometric embedding, it is easy to verify (using the triangular inequality) that S = {S q : q ∈ Q + } is a regular family. By Lemma 3.4 d R ([S], i R (x)) ≤ q, and thus d(ϕ([S]), ϕ(i R (x))) ≤ q, as ϕ is an isometric embedding. By the commutativity of the diagram, we have that d(ϕ([S]), j(x)) ≤ q. Since x ∈ S q we have also d(j(x), s) ≤ q, so by triangular inequality on X we conclude d(ϕ([S]), s) ≤ 2q for any q ∈ Q + , that is, ϕ([S]) = s and ϕ is onto.
(ii) Let j : X → X be the natural map, ϕ : X R → X the map defined in part (i) and i R : X R → X R the natural map. By Theorem 3.2, there is an isometric embedding ψ : X R → X such that the following diagram commutes Since j, ϕ are onto maps, then as the diagram commutes i R is also onto.
Richman [8] implicitly introduced a notion of maximal regular family by saying that a regular family {T q } q is maximal if for any other equivalent regular family {T q } q we have T q ⊆ T q for each q ∈ Q + . For instance, in Q with the usual premetric, the regular family {T q } q with T q = {q} is equivalent to the maximal regular family {S q } q with S q = {t ∈ Q : |t| ≤ q}. He observed that every equivalence class contains a maximal element which he called its canonical representative. The regular family given by (1) is maximal. Moreover, it is also a Cauchy family. It follows from the proof of the previous theorem that if T is a regular family and F is a Cauchy family such that la Vicerrectoría de Investigación y Extensión de la Universidad Industrial de Santander for the financial support for this work, which is part of the VIE project #2422.