International Journal of Mathematics and Mathematical Sciences

Volume 2013 (2013), Article ID 278381, 12 pages

http://dx.doi.org/10.1155/2013/278381

Research Article

A Solution to the Completion Problem for Quasi-Pseudometric Spaces

Athanasios Andrikopoulos

Department of Economics, University of Ioannina, P.O. Box 1186, 45110 Ioannina, Greece

Received 6 September 2013; Accepted 17 November 2013

Academic Editor: R. Lowen

Copyright © 2013 Athanasios Andrikopoulos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The different notions of Cauchy sequence and completeness proposed in the literature for quasi-pseudometric spaces do not provide a satisfactory theory of completeness and completion for all quasi-pseudometric spaces. In this paper, we introduce a notion of completeness which is classical in the sense that it is made up of equivalence classes of Cauchy sequences and constructs a completion for any given quasi-pseudometric space. This new completion theory extends the existing completion theory for metric spaces and satisfies the requirements posed by Doitchinov for a nice theory of completeness.

1. Introduction

A quasi-pseudometric space is a set together with a nonnegative real-valued function (called a quasi-pseudometric) such that, for every , (i) and (ii) . If satisfies the additional condition (iii) which implies that , then is called a quasimetric on . A quasimetric is a metric provided that . The conjugate of a quasi-pseudometric on is the quasi-pseudometric given by . By we denote the (pseudo)metric given by . Each quasi-pseudometric on induces a topology on which has as a base the family of -balls , where . A quasi-pseudometric space is if its associated topology is . In that case axiom (i) and the -condition can be replaced by () , . It is well known that every metric space has an (up to isometry) unique metric completion. For quasi-pseudometric spaces, the notion of completeness presents serious difficulties. The problem has been approached by several authors in [1–4], and so forth, but the solutions proposed have not solved these difficulties. Since a quasi-pseudometric space is a generalization of a metric space (the difference between pseudometrics and metrics is purely topological. In fact a pseudometric on is a metric if and only if the topology it generates is ), any completion theory for quasi-pseudometric spaces should generalize the usual completion theory for metric spaces. Traditionally this is done by generalizing the concept of Cauchy sequence and that of the convergence of a sequence. The appropriate generalization of the notion of Cauchy sequence in quasi-pseudometric spaces is no longer obvious. There are various generalizations to the notion of Cauchy sequence, but, up to now, none of these generalizations is able to give a satisfying completion theory for all quasi-pseudometric spaces. In fact, there exist many different notions of quasi-pseudometric completeness in the literature. More precisely, this problem has been studied in [3], where seven different notions of Cauchy sequence are presented. (The definitions are that of left and right  -Cauchy sequence ( is the quasimetric) defined by Reilly [5] and Subrahmanyam [6], respectively, -Cauchy sequence defined by Tan [7], right  - and left  -Cauchy sequence defined by Kelly [2], and weakly  right  - and weakly  left  -Cauchy sequence defined in [3].) By combining the seven notions of Cauchy sequences with the topologies , , and , we may reach a total of fourteen different definitions of “complete space” (considering the symmetry of using the instead of ). Since the concept of Cauchy sequence generalizes the concept of convergent sequence, it would be desirable that the notion of Cauchy sequence in any quasi-pseudometric space has to be defined in such a manner that () every convergent sequence is a Cauchy sequence. However, only one of the definitions presented in [3] satisfies the requirement . This definition is the following: a sequence in a quasi-pseudometric space is called Cauchy sequence, if for each there are a and a such that when . Unfortunately, this definition has two serious disadvantages. (i) The property of Cauchyness of a sequence depends not only on its terms but also on some other points which need not to belong to it (see [1, Example 2] and [2, page 88]). This allows sometimes for a convergent Cauchy sequence to cease Cauchyness if we remove the limit point from the space. (ii) Consider the real line equipped with the quasimetric , if and if . The sequence is a Cauchy one in the sense of the above definition. However (in Doitchinov’s words), “in view of the special character of the Sorgenfrey’s topology, it seems very inconvenient to regard this sequence as a potentially convergent one.”

Doitchinov in [1] introduced a class of quasimetric spaces, for which a satisfactory theory of completeness exists. According to Doitchinov, a notion of Cauchy sequence in any quasimetric space has to be defined in such a manner that this definition provides the properties that convergent sequences are Cauchy, and it agrees with the usual definition for metric spaces. Moreover, the suggested completion must be a monotone operator with respect to inclusion and give rise to the usual pseudometric completion in the pseudometric case. By means of Cauchy sequences which satisfy the above requirements, another concept of Cauchy sequence is proposed and a completion of a quasimetric space is constructed. However, this construction satisfies natural requirements only for balanced quasimetric spaces. (A quasimetric is called balanced if satisfies the following condition: If and are two sequence in and , then from for each , for each , and it follows that .) Unfortunately, Doitchinov’s condition of balancedness turns out to be rather restrictive. For instance, balanced quasimetrics induce completely regular Hausdorff topologies (see [1, Corollary 3] and [8, Page 208]) and totally bounded balanced quasimetrics induce uniformities (see [9, 10]). Künzi and Kivuvu [11, 12] localize Doitchinov’s idea of balancedness; that is, they do not work with arbitrary Cauchy filter pairs but only with those that they call balanced Cauchy filter pairs. Before stating the proposed completion theory from Künzi and Kivuvu, we give some definitions from [11]. Let be a -quasimetric space. It is known that can be isometrically embedded into a -quasimetric space , where the associated metric space is (isometric to) the completion of the metric space . We recall that is called bicomplete provided that each -Cauchy filter converges in . Let be nonempty subsets of . The -diameter from to is defined by . We will say that a pair of filters and on is a Cauchy filter pair on if . An arbitrary Cauchy filter pair on is said to converge to provided that and . A Cauchy filter pair on is said to be balanced on if for each we have . The space is called -complete provided that each balanced Cauchy filter pair converges in . Let be the set of all balanced Cauchy filter pairs on and let . Then, the formula ,   defines the distance from to . The space is a quasi-pseudometric space. We define an equivalence relation on the quasi-pseudometric space as follows: ,   if and only if ,   and ,  . Let be the set of all equivalence classes , where , with respect to . Then, on defined by ,   whenever determines a -quasimetric on . According to [11, Lemma 5] we can identify with the subspace of all balanced Cauchy filter pairs on that are minimal elements in the space of all balanced Cauchy filter pairs on . Künzi and Kivuvu in [11, Theorem 2] prove that if is a quasi-pseudometric space then is -complete. The space will be called the standard   -completion of . The authors also prove that the constructed standard -completion extends the bicompletion of every -quasimetric space. In contrast to -completion, the -completion can be strictly larger than the bicompletion in totally bounded -quasimetric spaces (see [12, Example 8]). But, sometimes, the -completion coincides with the bicompletion (see [11, Example 4]). In the present paper, we develop a nonsymmetric completion theory which gives a solution to the problem and extends naturally the existing completion theory for metric spaces. The following example shows that the proposed completion differs from the -completion.

Example 1 (see [2, Example 5.4]). Let . For each , let

It is easy to check that is a -quasi-pseudometric space. Let be the filter pair on generated by . We have that ) is a Cauchy filter pair on , which is not balanced, since

Therefore, the -completion of does not add new points to . In the proposed completion theory, the pair is a -cut in (see Definition 8). This -cut defines a new point in a complete quasi-pseudometric space (see Theorem 31). If the mapping is the isometric embedding (a map between quasi-pseudometric spaces is called an isometric embedding if for all and it is called an isometry if it is a surjective isometric embedding) such that is a dense subspace of , then and converge to with respect to and , respectively.

In this paper, (i) we propose a solution to the problem of quasi-pseudometric completion, called -completion; and (ii) we show that the proposed definition of Cauchy sequence and the constructed completion satisfy the requirements posed by Doitchinov for a satisfactory theory of completion of quasi-pseudometric spaces. To do that, we introduce the notion of -cut which, in a sense, is a generalization of the notion of the equivalence class of Cauchy sequences in metric spaces. This notion is inspired by the notion of the MacNeille-Dedekind’s cut [13] (see also [14]). In order to construct a -cut, we make use of pairs “sequence-cosequence” as in the Doitchinov’s quasimetric completion. The notion of -cut (a) allows us to restrict the “area” where a new point must stand and (b) enables us to construct the -completion by using Cauchy sequences instead of -cuts.

2. κ-Cauchy Cut

Throughout the paper will be an arbitrary quasi-pseudometric space, except the cases when it is explicitly stated that the space is . According to Doitchinov a sequence is called a cosequence to , if for any there are such that when and . In this case, we write or . Generally, given two sequences and in , we write if for any there are such that when and .

Definition 2 (see [3]). A sequence of a quasi-pseudometric space is called right (resp. left) - Cauchy, if for any there is an such that for each (resp. ). One calls (resp. ) the extreme index (resp. extreme point) of for .

Definition 3. Let be a quasi-pseudometric space and let , be two sequences on it. One says that is right (resp. left) -cofinal to  , if for each there exists satisfying the following property: for each there exists such that (resp. ) whenever . The sequences and are right (resp. left) -cofinal if is right (resp. left) -cofinal to and vice versa.

Proposition 4. Let be a right -Cauchy (left -Cauchy) sequence in a quasi-pseudometric space with a subsequence . Then, and are right (resp. left) -cofinal.

Proof. Let and be as in the statement of the proposition. Let be given. We know that from some index onwards we have that implies that . Now if we can find such that (as the form a strictly increasing sequence of integers). Hence, for each we have . Hence, is right -cofinal to . The converse is similar.

Proposition 5. In every quasi-pseudometric space , two right -cofinal right -Cauchy sequences have the same cosequences.

Proof. Let and be two right -cofinal sequences. Suppose that is a cosequence of . Fix an . Then there exists , such that for and . On the other hand, there is with the following property: for each there exists such that whenever . If , then for each , . Hence, is a cosequence of .

Similarly we can prove the following proposition.

Proposition 6. In every quasi-pseudometric space , two left -cofinal left -Cauchy sequences are cosequence of the same sequences.

The two previous propositions imply the following corollary.

Corollary 7. In every quasi-pseudometric space , two right (left) -cofinal sequences have the same limit points for (resp. ).

In the following definition, without loss of generality, we may suppose that, for , it is .

Definition 8. Let be a quasi-pseudometric space. We call -cut in an ordered pair of families of right -Cauchy sequences and left -Cauchy sequences, respectively, with the following properties.

(i)For any and any there holds

(ii)Any two members of the family (resp. ) are right (resp. left) -cofinal.(iii)The classes and are maximal with respect to set inclusion.

We call (resp.   first  (resp. second) class of . In what follows, for simplicity of the proofs, we call the elements of left -Cauchy consequences of the elements of .

Definition 9. To every one chooses a -cut , where consists of right -Cauchy sequences which converges to with respect to and consists of left -Cauchy cosequences which converges to with respect to . The sequence itself belongs to both of the classes. If there are not right -Cauchy sequences (resp. left -Cauchy cosequences) converging to , then, (resp. ). If is an isolated point for and , respectively, then .

As we can map to the set of all -cuts, it is possible that for some there correspond many -cuts; let , (the index set depends on the choice of ). Then, we define

Without loss of generality, we assume that for some .

Remark 10. If the space is , then the function defined above is an injective function (one-to-one) of into . Indeed, let be such that . Then, . Thus, which implies that .

Definition 11. One calls -Cauchy sequence any right -Cauchy sequence which is member of the first class of a -cut.

Definition 12. Two right -Cauchy sequences and defined in a quasi-pseudometric space are called -equivalent if every left -Cauchy cosequence to is a left -Cauchy cosequence to and vice versa.

Clearly, -equivalence defines an equivalence relation on .

An immediate consequence of Definitions 8, 11, and 12 is the following corollary.

Corollary 13. In every quasi-pseudometric space , two nonconstant -Cauchy sequences which belong to the same -cut are -equivalent.

Remark 14. In view of Corollary 13, we have that the first class of a -cut is an equivalence class of the -Cauchy sequences that are considered to be equivalent by -equivalence relation.

Definition 15. A quasi-pseudometric space is called -complete if every -Cauchy sequence converges.

Definition 16. A -completion of a quasi-pseudometric is a -complete quasi-pseudometric space in which can be isometrically embedded as a dense subspace.

If a -cut does not belong to we say that is a -gap. The set of all -gaps of is symbolized by . Let us define

Throughout the paper, for every , , denote the two classes of . In this case, we write .

Definition 17. Let be a quasi-pseudometric space. Suppose that is a nonnegative real number, , , and . We put if

(i) or(ii)for each there are , such that

when and . If for some , then the arbitrary sequence always coincides with the fixed sequence, for which for all . That is, if

when .

Then we let

Proposition 18. Let , and . Suppose that . Then, is right -cofinal to .

Proof. By the suppositions, for each there are , such that whenever and . The rest is evident by Definition 3.

Proposition 19. The truth of in Definition 17 depends only on , , and ; it does not depend on the choice of the sequences and .

Proof. Let , , , and be as in the supposition of Definition 17. Further, let . Then, for each there are , such that

when and .

We choose two arbitrary sequences and . Since is right -cofinal to , there is satisfying the following property: for each there exists such that

whenever . Let . Then, we have

whenever and .

Analogously, since is right -cofinal to , there is satisfying the following property: for each there exists such that

whenever . Let . Then, by (11) we have

when and .

Proposition 20. Let , , and . Then,

Proof. Let . Then, for any there are , such that

when and . To prove that for and , suppose to the contrary that there exist a subsequence of and a subsequence of such that for all there holds . Then, since Propositions 4, 5, and 6 imply that and , it follows, according to Definition 17, that , a contradiction. Therefore, we have .

Proposition 21. is a quasi-pseudometric.

Proof. From Definition 17 it follows immediately that and for all . To prove that satisfies the triangle inequality let . We distinguish the following cases.

(i) and . Suppose that , , , , and . Then, for any there are , such that whenever and . Similarly, there are , such that whenever and . Let . Then, for each , . Hence, according to Definition 17, we have

(ii) and . Suppose that . Since , Definition 17 implies that and . Hence

(iii) and . This case is proved similar to the previous one.(iv) and . This case is trivial.

After the definition of the quasi-pseudometric , we are going to show that the function of Definition 9 is an isometric embedding of into .

Proposition 22. For any there holds .

Proof. The assertion of the proposition follows from the fact that , , and Proposition 20.

Proposition 23. For any ,

(a)if then ;(b)if then .

Proof. (a)  Let . Then, there exists such that whenever . Now pick a right -Cauchy sequence of . Since is right -cofinal to , there is satisfying the following property: for each there exists such that

whenever . Suppose that . Fix an and let . Then, and so there exists an such that for each . Hence,

for and . Since and is an arbitrary right -Cauchy sequence of , we conclude that . Hence, whenever .

(b) The proof is similar to that of (a).

Proposition 23 implies the following corollary.

Corollary 24. The set is dense in .

Proposition 25. Let be a nonconstant right -Cauchy sequence of without last element. Then there exists a right -Cauchy sequence of such that the sequences and are right -cofinal right -Cauchy sequences.

Proof.

(a)  The Construction of the Right  -Cauchy Sequence    of  . Let be a -Cauchy sequence in . Without loss of generality we can assume that, for each , . Let be a nonzero fixed natural number and suppose that is the smallest integer with the property

Then, there holds

For each , we also fix a sequence

If for some , then for each . Let

Then, for each we have

Hence, since , there are and such that

whenever and . On the other hand, since is a right -Cauchy sequence, there exists an such that

for each .

We put . Then, we have

Clearly, (25) and (26) are valid if we replace and , respectively, by . Let be such that . Then, since , similarly to the above we get

whenever and for some . Let . Then, by putting in (25), , and , we obtain that

On the other hand (28) implies that

for each for some . Hence,

There are two cases to consider: (i) ; (ii) .

In case (i), (28) implies that

Hence,

In case (ii), we have ; hence (26) implies that

Therefore, (29), (30), and (34) imply that

Hence, in any case, we have

for each .

We consider the set

According to (27) we give to the following order:

if and only if

Since is countable and is a linear order, without loss of generality, we may assume that and , where is the standard ordering of the natural numbers (this means that is a linear refinement of ). We now prove that the sequence

is a right -Cauchy sequence which satisfies the assumptions of the lemma.

Let and . Suppose that

We prove that

If and , then . Therefore,

Suppose that . Then, we have two cases to consider: (I) and ; (II) . In case (I), we have . Since is a right -Cauchy sequence we have that

We proceed to prove case (II). In this case we have two subcases, and . In subcase we have . Since is a right -Cauchy sequence we have that

Then, (36) and (45) imply that

In order to prove subcase , we combine (29) and (30). So, we get

for each for some . Now, we have two subcases to consider: the first subcase is that and the second one is that ,. In the first subcase, (45) implies that

In the second subcase, we have

Therefore, (45) and (49) imply that

Hence, in any case the inequality (42) is invalid.

(b)    and    Are Right  -Cofinal. We first show that is right -cofinal to . Fix an and let . Suppose that . Then, we define , where .

We prove that

for each . That is, if , then

for each for some . Indeed, fix a such that . Then, from (25) which is valid for we have

for each . Now, since we have

for some , onwards. On the other hand, since is right -Cauchy sequence we have

for . Finally, by combining the last three inequalities, we obtain

for each . The last conclusion confirms the truth of (51).

To prove that is right -cofinal to , fix an . We define . Now, let and let . We prove that

for each . Indeed, let and . Since

using the same logic as above, we conclude that

for each for some . On the other hand, since we have

whenever for some . Therefore, (59) and (60) imply that

for each and each . Finally, (61) implies (57) and therefore the required sequence of the hypothesis is the sequence .

Proposition 26. Let be a nonconstant left -Cauchy sequence of without last element. Then there exists a left -Cauchy sequence of such that the sequences and are left -cofinal.

Theorem 27. Every quasi-pseudometric space has a -completion.

Proof. Let be a quasi-pseudometric space and let be a -Cauchy sequence in the space . Then, by Definition 11, there exists a -cut such that . Suppose that and . Then, for some . We prove that there exists a -cut in such that converges to with respect to .

We define that is a right -Cauchy sequence of such that and is a left -Cauchy sequence of such that .

We first verify that constitute a -cut in . For this we need to show that

(A)the classes , are nonvoid,(B)the pair satisfies the conditions of Definition 8.

We first show (A). We have two cases to consider.

Case A.1. There exists    such that the sequence    is nonconstant. In this case, Proposition 25 implies that there is a right -Cauchy sequence of such that and are right -cofinal. Therefore, which implies that .

Case A.2. For each  , there exists    such that    whenever  . Fix an . Suppose that . Then, from ,  , and , , we conclude that . It follows that . Therefore, in any case we have that . Similarly we prove that .

To show (B), we first prove the validity of Condition (i) of Definition 8. Let and . Then, by construction of and , we have . Hence, Proposition 22 implies that . To prove that satisfies the second condition of Definition 8, let and be two right -Cauchy sequences of . Since and belong to , it follows by definition that they are right -cofinal. Hence, Proposition 22 implies that and are right -cofinal. Clearly, the maximality of and implies the maximality of and , respectively. Hence, condition (iii) is satisfied.

It remains to prove that converges to with respect to . If is constant, then, for some onwards. Hence, by the Definition 17, we have that . If the sequence is nonconstant, then there exists a right -Cauchy sequence such that and are right -cofinal. Since , Proposition 23 implies that . So, by Corollary 7 we conclude that . This completes the proof.

Definition 28. Let be a -quasi-pseudometric space. We define an equivalence relation on as follows: if and only if whenever or and for some whenever , (see Definition 9). In the following will denote the isometric quotient map with respect to previously defined. That is, corresponds to whenever . Clearly, is surjective. will denote the set of all equivalence classes .

Proposition 29. Let be a -quasi-pseudometric space and let , be as above. Suppose that is a function mapping defined by whenever . Then, determines a -quasi-pseudometric on .

Proof. Clearly, is an equivalence relation on . To prove that is well-defined suppose that and and . By the triangle inequality we see that and hence . Similarly, which implies that . Hence, is well defined. It is obvious that is quasi-pseudometric. To prove that is a -quasi-pseudometric space suppose that . Then, . Suppose that . Let and . Then, by Proposition 18 we conclude that and are right -cofinal. Therefore, . If and for some , then which implies that . Since is we conclude that . Hence, in any case we have . Consequently, is a -quasi-pseudometric space.

Proposition 30. If is a -quasi-pseudometric space, then is an isometric embedding.

Proof. We have that is an isometry, since both and are isometries. On the other hand, for any , implies that . Then, Proposition 22 implies that . Hence, because is a -quasi-pseudometric space. Therefore, is injective which implies that is an isometric embedding.

Theorem 31. Let be a -quasi-pseudometric space. Then is a -completion of .

Proof. Let be a -Cauchy sequence in . Therefore, since is a surjective isometry, is a -Cauchy sequence in . Thus for some , converges to . It follows that converges to since is an isometry. So is -complete -quasi-pseudometric space (see Proposition 29). It remains to prove that the set is dense in . Indeed, suppose that . Then, since is a surjective isometry, there exists such that . Let . Then, by Definition 17 we have that . Therefore, converges to with respect to . It follows that is dense in which implies that is a -completion of (see Definition 16).

Definition 32. Let be a -quasi-pseudometric space. Then the -quasi-pseudometric space previously defined will be called the standard  -completion of .

3. A Categorical View of the κ-Completion

As already pointed out, a natural problem which arises in quasi-pseudometric spaces is to introduce a convenient notion of completeness and to construct a satisfactory theory of completion. What do we mean by a “satisfactory theory” of completeness and completion for quasi-pseudometric spaces? According to Doitchinov, the answer to that question is based on certain restrictions that we must have. These restrictions are separated in two groups as follows.

 (α)A notion of Cauchy sequence in any quasi-pseudometric space has to be defined in such a manner that the following requirements are fulfilled:

(i)every convergent sequence is a Cauchy sequence;(ii)when is a pseudometric space, the Cauchy sequences are the usual ones.

(β)A standard construction of a completion of any quasi-pseudometric space may need the following.

(iii)If , then , where inclusions are understood as quasimetric embeddings (an embedding between two objects and is an injective (one-to-one) and structure-preserving map . A structure on a set consists of additional mathematical objects (topological structures, orders, and equivalence relations) that in some manner attach to the set. The precise meaning of “structure-preserving” depends on the kind of mathematical structure of which and are instances. In general topology, an embedding is a homeomorphism or topological isomorphism (a bicontinuous function between two topological spaces) onto its image. More explicitly, a map between topological spaces and is an embedding if yields a homeomorphism between and (where carries the subspace topology inherited from ). In quasimetric spaces such homeomorphisms are called isometries or quasimetric embeddings.) and the second one is an extension of the former.(iv)In the case when is a metric space, is the usual metric completion of .

After the paper of Künzi and Schellekens [15] another requirement must be added in (β). This is the following.

(v)The constructed completion must be an idempotent operation; that is, the -completion of the -completion of is isomorphic to the -completion of .

The -completion differs with all other previously introduced completion theories for quasi-pseudometric spaces. More precisely, in metric spaces, symmetry enables us to construct the classical completion by using Cauchy sequences instead of equivalence classes of Cauchy sequences. Similarly in the -procedure, since right (resp. left) -cofinality between right -Cauchy sequences (resp. left -Cauchy cosequences) is an equivalence relation, we use Cauchy sequences instead of -cuts. On the other hand, in metric spaces, the notions of Cauchy sequence, right -Cauchy sequence, and left -Cauchy cosequence coincide. Then, in the -procedure, every class of equivalent Cauchy sequences of the classical case, identifies with the class of right -Cauchy sequences of a -cut and vice versa. We now show that the notion of -completeness and the constructed -completion satisfy the requirements posed by Doitchinov [1, page 128] as well as the requirement (v) previously posed.

The following two propositions show that the -completion satisfies the requirements (i) and (ii) previously stated.

Proposition 33. Every convergent right -Cauchy sequence is a -Cauchy sequence.

Proof. In fact, if a right -Cauchy sequence converges to a point of , then it belongs to the first class of a -cut . The validity of this proposition follows from Definition 11.

Clearly, in the case of metric spaces, the notions of converging right -Cauchy sequence and converging sequence coincide.

The following proposition is obvious.

Proposition 34. If is a pseudometric space, then the notion of -Cauchy sequence coincides with the classical one.

Remark 35. Doitchinov in [1, Proposition 1] shows that if a sequence converges to a point, say , then has as its cosequence, so it is a -Cauchy sequence. (A sequence in the quasimetric space is called -Cauchy sequence provided that for any there exists a and such that when .) In another point in [1, page 130], he also says that a “good” -Cauchy sequence is that which remains -Cauchy sequence even if we withdraw the limit point from the space. We recall his counterexample as follows: let be a quasimetric space, and

The sequence converges to , so it must be a -Cauchy sequence. However, if we withdraw the limit point , the space becomes a discrete metric space, so ceases to be a -Cauchy sequence. Of course, this conclusion contradicts with Doitchinov’s first aspect. It is easy to see that with the notion of -Cauchy sequence we avoid this undesirable situation. In fact, is not a -Cauchy sequence.

The requirement (iii) of Doitchinov concerns the categorical aspect of a completion. It is well known that the class of -complete quasimetric spaces (a quasimetric space is called -complete if every -Cauchy sequence is convergent) defines a category provided that we assume as morphisms the quasiuniformly continuous functions. Bonsangue et al. [16] defines the category with complete generalized metric spaces (a generalized metric space consists of a set together with a distance function , satisfying and , for all , and in . We remark that traditionally a quasi-pseudometric space is required to take finite values. However the Yoneda completion of [16] involves generalized metric which allows for infinite distances) as objects and nonexpansive maps (a function between two quasi-pseudometric spaces and is non-expansive provided that for all .) as morphisms.

We define a new category whose objects are the quasi-pseudo-metric spaces and morphisms are the -cut preserving functions defined below (Definition 39).

Definition 36. A function between two quasi-pseudometric spaces and is called quasiuniformly continuous if for any there is a such that whenever .

Definition 37. Two sequences and in a quasi-pseudometric space form a -Cauchy pair of sequences if there exists a -cut in such that and . Two -Cauchy pairs of sequences are -compatible if they belong to the same -cut.

Definition 38. Let , be quasi-pseudometric spaces and let be a continuous function. Let also , be a -Cauchy pair of sequences in . We say that a -Cauchy pair , of sequences in is -compatible to , if it satisfies the following conditions:

(i) and are -compatible;(ii), is a -Cauchy pair of sequences in .

Definition 39. Let , be quasi-pseudometric spaces. A continuous function is called -cut preserving if it satisfies the following conditions.

(i)For each -Cauchy pair of sequences in , there exists a -Cauchy pair of sequences in such that is -compatible to .(ii)If , and are -Cauchy pairs of sequences such that is -compatible to and is -compatible to , then, the -compatibility (resp. non--compatibility) of and implies the -compatibility (non--compatibility) of and , respectively.

By the previously mentioned definition, it is clear that a cut preserving function takes distinct elements of to distinct elements of . On the other hand, a cut preserving cut is one that “embeds” as a subset of . In view of the Definitions 8, 11, and 37, Proposition 22, and Theorem 27, one obtains that the quasi-pseudometric embedding is a -cut preserving function.

Doitchinov [1] uses quasiuniformly continuous functions defined in quasimetric spaces. Lemma  7 in [1] shows that these functions are -cut preserving functions.

Similarly to what is in the theory of metric spaces, the categorical notion of a quasi-pseudometric completion is the following: a complete quasi-pseudometric space is called a -completion of a given quasi-pseudometric space if there exists a -cut preserving function such that

(a) is dense in ;(b)the quasi-pseudometric is the inverse image of under ;(c)for any -cut preserving function , where is a -complete quasi-pseudometric space, there is a -cut preserving function with .

The mapping is called the canonical quasi-pseudometric embedding of into .

Proposition 40. Let , be two quasi-pseudometric spaces, let be -complete, and let be a -cut preserving function. Then, there exists a -cut preserving function such that .

Proof. Let , , and be as in the hypothesis of the proposition. Now define a mapping by letting whenever is a -cut in such that and . Definitions 28 and 39 imply that takes distinct elements of to distinct elements of . Let where be the isometric embedding defined in Proposition 30. Then, by the continuity of at , we have . It remains to prove that is -cut preserving. Let be a pair Cauchy of sequences in . Suppose that for each we have and . Then, let be a pair Cauchy of sequences in . Then, by Propositions 25 and 26 and Theorem 27 there is a pair Cauchy , of sequences in such that and , are -compatible, and are right -cofinal, and and are left -cofinal. Since is cut preserving, there exists a Cauchy pair , of sequences satisfying the conditions (i) and (ii) of Definition 39. Since is an isometry and it follows that satisfies the conditions (i) and (ii) of Definition 39. Hence, is -cut preserving.

The previous proposition shows that the standard -completion is the smallest -complete quasi-pseudometric space containing in the sense that any other -complete space containing as a subspace also contains as a subspace.

The next two propositions show that the -completion satisfies the requirements (iii) and (iv) posed by Doitchinov.

Proposition 41. Let , be two quasi-pseudometric spaces, and let , be their standard -completions with canonical maps and . If is a -cut preserving function, then there exists a -cut preserving function .

Proof. The existence of a -cut preserving function follows from Proposition 40 when one applies it to the function .

Proposition 42. If is a metric space, then the standard -completion coincides with the usual metric completion.

Proof. Let be a metric space. Then in the -procedure the notions of right -Cauchy sequence, left -Cauchy cosequence, and Cauchy sequence coincide. Hence, every class of equivalent Cauchy sequences of the classical case identifies with a -cut and vice versa. Hence, the new elements of the two completions, the first of the classical completion and the second of the -completion, coincide. The rest is obvious.

Finally, let us discuss the requirement (v) previously stated in the introduction of this section. It is well known in classical metric space theory that Cauchy completion is idempotent. One can also verify that the bicompletion as well as the Doitchinov completion is an idempotent operation. The idempotency of the well known completions for quasi-pseudometric spaces has not been studied. There are also two main types of domain theoretic completions, the Smyth completion [17] and the Yoneda completion [16]. The Smyth completion is idempotent, while the Yoneda completion is not (see [15, page 164]). Recently, Künzi and Schellekens in [15] introduce a general net-version for the Yoneda completion and characterize the family of generalized metric spaces (generalized metric spaces are a common generalization of preorders and ordinary metric spaces. Such spaces consist of a set together with a distance function satisfying and , for all and in .) for which the Yoneda completion is idempotent.

We recall the classical definition of idempotency (adapted to the specific case of the -completion). In the following denotes the -completion of the -completion .

Definition 43. The -completion of a quasi-pseudometric space is idempotent if and only if there exists an isometry such that for each , for some .

Proposition 44. The -completion is idempotent.

Proof. Let be a quasi-pseudometric space and let be its -completion. Let us denote by the set of -cuts in and let . Suppose that and . Then, the families and , where and , define a -cut in . Then, according to Theorem 27, to each -cut there corresponds a -cut . Let and let be the quasi-pseudometric embedding of into as it is defined in Definition 9. Then, . Therefore, we have . Next, for each we have

where and . Clearly, is an isometry. To complete the proof we must show that is -complete. Indeed, let be a -Cauchy sequence in . Then, there exists a -cut in such that . Suppose that , and for some . For each and each we have and . On the other hand, and for some . Next, we define , where and . By Theorem 27, there exists a -cut such that, for each , with respect to . By definition of , it follows that with respect to , where . Since is an isometry, Proposition 25 implies that and are right -cofinal. Therefore, Corollary 7 implies that converges to with respect to . It follows that converges to with respect to .

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