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International Journal of Mathematics and Mathematical Sciences
Volume 2013 (2013), Article ID 278381, 12 pages
A Solution to the Completion Problem for Quasi-Pseudometric Spaces
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.
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.
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 , 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  and Subrahmanyam , respectively, -Cauchy sequence defined by Tan , right - and left -Cauchy sequence defined by Kelly , and weakly right - and weakly left -Cauchy sequence defined in .) 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  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  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 . 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  (see also ). 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 ). 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 .
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.
(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