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Abstract and Applied Analysis
Volume 2014, Article ID 163258, 9 pages
http://dx.doi.org/10.1155/2014/163258
Research Article

Remarks on Ultrametrics and Metric-Preserving Functions

1Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom 73000, Thailand
2Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand

Received 22 December 2013; Accepted 19 April 2014; Published 30 April 2014

Academic Editor: Gaston M. N’Guérékata

Copyright © 2014 Prapanpong Pongsriiam and Imchit Termwuttipong. 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

Functions whose composition with every metric is a metric are said to be metric-preserving. In this paper, we investigate a variation of the concept of metric-preserving functions where metrics are replaced by ultrametrics.

1. Introduction

Under what conditions on a function is it the case that for every metric space , is still a metric? It is well known that for any metric , and are bounded metrics topologically equivalent to , while need not be a metric.

We call metric-preserving if for all metric spaces , is a metric. Therefore, the functions and given by and are metric-preserving but is not. The concept of metric-preserving functions first appears in Wilson’s article [1] and is thoroughly investigated by many authors; see for example, [218] and references therein.

However, other important types of distances such as ultrametrics, pseudometrics, pseudodistances [19, 20], -distances, and -distances have not yet been developed in the connection with metric-preserving functions. These distances have many applications in mathematics; see, for example, applications of -distances and -distances in [2127]. We will particularly be concerned with the ultrametrics which arise naturally in the study of -adic numbers and nonarchimedean analysis [28, 29], topology and dynamical system [3033], topological algebra [34], and theoretical computer science [35].

In connection with ultrametrics and metric-preserving functions, the problem arises to investigate the properties of the following functions and compare them with those of metric-preserving functions.

Definition 1. Let . We say that (i)is ultrametric-preserving if for all ultrametric spaces , is an ultrametric;(ii) is metric-ultrametric-preserving if for all metric spaces , is an ultrametric;(iii) is ultrametric-metric-preserving if for all ultrametric spaces , is a metric.For convenience, we also let be the set of all metric-preserving functions, the set of all ultrametric-preserving functions, the set of all ultrametric-metric-preserving functions, and the set of all metric-ultrametric-preserving functions.

We will give some basic definitions and useful results that will be used throughout this paper in the next section. We then give properties and characterizations of those functions in Sections 3, 4, and 5. We discuss and give some results on the continuity aspect of those functions in Section 6.

2. Preliminaries and Lemmas

In this section, we give some basic definitions and results for the convenience of the reader. First, we recall the definition of a metric space and an ultrametric space.

A metric space is a set together with a function satisfying the following three conditions:(M1) For all , if and only if ,(M2) for all , , and(M3) for all , .An ultrametric space is a metric space satisfying the stronger inequality (called the ultrametric inequality): (U3) for all , .A metric space is said to be topologically discrete if for every there is an such that , where denote the open ball center at and radius . In addition, is said to be uniformly discrete if there exists an such that for every .

Next we recall the definitions concerning certain behaviors of functions. Throughout, we let and let . Then is said to be increasing on if for all satisfying , and is said to be strictly increasing on if for all satisfying . The notion of decreasing or strictly decreasing functions is defined similarly.

The function is said to be amenable if , and is said to be tightly bounded on if there is such that for all . We say that is subadditive if for all , is convex if for all and , and is concave if for all and . As mentioned earlier, we say that is metric-preserving if for all metric spaces , is a metric. Furthermore, is strongly metric-preserving if is a metric equivalent to for every metric .

Now we are ready to state the results which will be applied in the proof of our theorems.

Lemma 2. Let . If is amenable, subadditive, and increasing on , then is metric-preserving.

Proof. The proof can be found, for example, in [4, 6].

Lemma 3. If is amenable and tightly bounded, then is metric-preserving.

Proof. The proof can be found, for example, in [3, 4].

The next lemma might be less well known, so we give a proof here for completeness.

Lemma 4. If is amenable and concave, then the function is decreasing on .

Proof. Let and . Since is concave, we obtain Therefore, , as desired.

Lemma 5. Let be an ultrametric space. Then for every ,

Proof. We have A repeated application of the ultrametric inequality as above gives the desired result.

Next we give basic relations and properties of the functions in , , , and .

Proposition 6. The following relations hold , .

Proof. Since an ultrametric is a metric, and . So follows. Similarly, and , so holds. and are true in general.

We will obtain characterization of the functions in , , and in later section. Then we will show that the relation in Proposition 6 is in fact a proper subset. It is easy to see that if , then is amenable. We extend this to the case of any function .

Proposition 7. If , then is amenable.

Proof. Assume that . To show that is amenable, we let be such that . Let , where , , and . Let be the Euclidean metric on and let be the restriction of on . Then . Therefore, is an ultrametric space. So is a metric on . Now , and , which implies . That is, . Hence . This shows that is amenable as desired.

Corollary 8. If is in , , , or , then is amenable.

Proof. By Proposition 6, . So the result follows from Proposition 7.

3. Ultrametric-Preserving Functions

In this section, we obtain characterizations of ultrametric-preserving functions. Then we compare their properties with those of metric-preserving functions.

Theorem 9. Let . Then is ultrametric-preserving if and only if is amenable and increasing.

Proof. Assume that is ultrametric-preserving. By Corollary 8, it suffices to show that is increasing. Let and . Let be the Euclidean metric on and let , where , , and . Let be the restriction of on . Then , . Therefore, is an ultrametric space. Since is ultrametric-preserving, is an ultrametric. Therefore, as required. Next assume that is increasing and amenable. Let be an ultrametric space, and let . Since is amenable, if and only if . Since is an ultrametric, . So or . If , then . If , then . In any case . Therefore, is an ultrametric. This completes the proof.

Corollary 10. Let . Then the following statements hold: (i)if is ultrametric-preserving and subadditive, then is metric-preserving;(ii)if is metric-preserving and increasing on , then is ultrametric-preserving.

Proof. We obtain that (i) follows from Theorem 9 and Lemma 2, and (ii) follows from Corollary 8 and Theorem 9.

The next example shows that and .

Example 11. Let be given by

By Theorem 9, is ultrametric-preserving and is not ultrametric-preserving. If is the usual metric on , we see that So is not a metric and therefore is not metric-preserving. Since for all , is tightly bounded, and therefore, by Lemma 3, is metric-preserving. In conclusion, , , , and . This shows that and . This example also shows that the relations and in Proposition 6 are proper subsets.

Next we give some results concerning concavity of the functions in .

Theorem 12. Let . If is amenable and concave, then is ultrametric-preserving.

Proof. Assume that is amenable and concave. We will show that is increasing. First observe that if , then because is amenable. Next let and suppose for a contradiction that . Let , , and . Then , and . Since is concave, we obtain This implies that which contradicts the fact that and is amenable. Hence is increasing on . By Theorem 9, is ultrametric-preserving.

Corollary 13. If is amenable and concave, then is both ultrametric-preserving and metric-preserving.

Proof. The first part comes from Theorem 12. The other part has appeared in the literature but we will give an alternative proof here. We know that is increasing by Theorems 12 and 9. So by Lemma 2, it suffices to show that is subadditive. Let . By Lemma 4, we have . Therefore, as required. This completes the proof.

The next example shows that there exists a function which is both metric-preserving and ultrametric-preserving but not concave.

Example 14. Let be defined by

It is easy to see that is amenable and increasing. So, by Theorem 9, is ultrametric-preserving. Next we will show that is metric-preserving. By Lemma 2, it suffices to show that is subadditive. Observe that and for every . We consider in several cases.If , then .If , then .Similarly, if , then .If , , then If , , then .If , , then .If , , then .The other cases can be obtained similarly. Therefore, is subadditive. Hence, is metric-preserving. But , so is not concave. That is, but is not concave. In addition, is not a constant on . So this example also shows that and the relation in Proposition 6 is a proper subset.

4. Metric-Ultrametric-Preserving Functions

In this section, we characterize the functions in . We will see that this notion is so strong that it forces the functions to be a constant on . More precisely, we obtain the following theorem.

Theorem 15. Let . Then is metric-ultrametric-preserving if and only if is amenable and is a constant on .

Proof. First assume that is amenable and is a constant on . That is there exists a constant such that To show that is metric-ultrametric-preserving, let be a metric space and let . If or or , then it is easy to see that . If are all distinct, then and therefore This shows that is an ultrametric. In the other direction, we assume that . By Corollary 8, it is enough to show that is a constant on . Throughout the proof, we let be the usual metric on and the Euclidean metric on . We will apply Lemma 5 repeatedly. First we will show that So we let be arbitrary. Since , is an ultrametric on . By Lemma 5, we have Next let , , be points in . Since , is an ultrametric on . Therefore, Therefore, . By a similar method, we obtain In addition, we let , , be points in so that Therefore, . Hence for every , as asserted. We conclude that Next let . We will show that . Let be such that . Let , , , , , be points in . By (18) and the fact that is an ultrametric on , we obtain This shows that From (18) and (20), we see that for all . This completes the proof.

Let be a metric-preserving function and let be a metric. Then either is a metric equivalent to or is a uniformly discrete metric [3, 6]. In addition, is continuous on if and only if it is continuous at 0 [3, 4, 6]. But by Theorem 15, every metric-ultrametric-preserving function is always discontinuous at 0 and is always a uniformly discrete metric for all metric . We record this in the next corollary.

Corollary 16. Let be metric-ultrametric-preserving. Then (i) is a uniformly discrete metric for every metric ,(ii) is discontinuous at and is continuous on .

Proof. By Theorem 15, there exists such that So (ii) follows immediately. If is a metric space, then So if we let , then for every . This proves (i).

5. Ultrametric-Metric-Preserving Functions

In this section, we give a characterization of the functions in in terms of special type of triangle triplets. Recall that a triple of nonnegative real numbers is called triangle triplet if , , and . We denote by the set of all triangle triplets. We introduce a special type of triangle triplets that will be used to characterize ultrametric-metric-preserving functions in the next definition.

Definition 17. A triple of nonnegative real numbers will be called ultra-triangle triplet if , , and . We denote by the set of all ultra-triangle triplets.

Since we will compare the functions in with those in , we first state a characterization of metric-preserving functions in terms of triangle triplets.

Theorem 18. Let be amenable. Then the following statements are equivalent: (i) is metric-preserving,(ii)for each , ,(iii)for each , .

Proof. The proof can be found, for example, in [3, 4, 6].

Similar to Theorem 18, we obtain a characterization of the functions in in terms of ultra-triangle triplets as follows.

Theorem 19. Let be amenable. Then the following statements are equivalent: (i) is ultrametric-metric-preserving,(ii)for each , ,(iii)for each , .

To prove Theorem 19, the following lemmas are useful.

Lemma 20. If is an ultrametric space and , then the triple is an ultra-triangle triplet. Conversely, if is an ultra-triangle triplet, then there exist an ultrametric space and such that .

Lemma 21. If , then

We will prove Lemmas 21 and 20, and then Theorem 19, respectively.

Proof of Lemma 21. Let . Suppose that are all distinct. Without loss of generality, we can assume that . Then which contradicts the fact that . So are not all distinct. If , then and (iii) holds. Similarly, if , then (ii) holds and if , then (i) holds.

Proof of Lemma 20. The first part follows immediately from the ultrametric inequality of . For the converse, we let . By Lemma 21, we can assume that (the other cases can be proved similarly). Let , where , , and . Let be the Euclidean metric on and . Then is an ultrametric space and .

Proof of Theorem 19. (i)(ii) Let and let . Then by Lemma 20, there exist an ultrametric space and such that Since , is a metric space. It follows from the triangle inequality of that is a triangle triplet. That is, .
(ii)(iii) Assume that (ii) holds. Let . Then, . So by (ii). Therefore, , as required.
(iii)(i) Assume that (iii) holds. Let be an ultrametric space. Since is amenable, if and only if . So it remains to show that the triangle inequality holds for . Let . Then by Lemma 20, . Then by Lemma 21, we can assume that (the other cases can be proved similarly). Then by (iii), we obtain Hence, the proof is complete.

Next we give an example to show that the relation in Proposition 6 is a proper subset.

Example 22. Let be given by

Let be the usual metric on . Then So is not a metric and therefore . Since is not increasing, . Next we will show that , by applying Theorem 19. Let . If , then and therefore for all . In particular, . If , then and thus . In any case, we have . Hence but and . This example shows that and the relation in Proposition 6 is in fact a proper subset.

Remark 23. From Examples 11, 14, and 22, we now see that the relations , , , and in Proposition 6 are in fact proper subsets.
If we replace in the definition of in Example 22 by a constant (that is, if and if ), then if and only if .

6. Continuity

In this section, we investigate the continuity aspect of the functions in , , , and . By Corollary 16, the continuity of metric-ultrametric-preserving functions is trivial: they are always discontinuous at and continuous elsewhere. The continuity of metric-preserving functions has also been investigated by many authors [14, 6, 8, 18], but we can still extend it further in the next theorem.

Before we state the theorem, let us recall some definitions concerning generalized continuities. Let . Then is said to be weakly continuous at if and only if there are sequences and such that is strictly increasing and converges to , is strictly decreasing and converges to , and and converge to . If , then is said to be weakly continuous at if and only if there exists a strictly decreasing sequence converging to such that converges to . We refer the reader to [36] for weak continuity of functions defined on a more general domain.

Unlike weak continuity, quasi continuity and almost continuity seem to be first given in a more general domain than a subset of . So we let and be topological spaces and let . Then is said to be quasi continuous at if for all open sets of and of such that and , there is a nonempty open sets of such that and . The function is said to be almost continuous at in the sense of Singal (briefly a.c.S. at ) if for each open set of   containing , there exists an open set containing such that and is said to be almost continuous at in the sense of Husain (briefly a.c.H. at ) if for each open set of containing , is a neighborhood of . The function is said to be quasi continuous on (or a.c.S. on , or a.c.H. on ) if it is quasi continuous at every (a.c.S. at for every , a.c.H. at for every ).

Remark 24. The concepts of a.c.S. functions and a.c.H. functions are not equivalent as shown by Long and Carnahan [37].
There are several other types of continuities in the literature. Some of them have the same name but different definition, see [38] for instance, a different definition of weak continuity. We refer the reader to [3943] and the other references for additional details and information.

Now we are ready to state our theorem. We will see that there is a similarity and dissimilarity between continuity of the functions in and .

Theorem 25. Let be metric-preserving. The following statements are equivalent: (1) is continuous at ,(2) is continuous at ,(3)For every , there exists and such that ,(4) is strongly metric-preserving,(5) is uniformly continuous on ,(6) is weakly continuous on ,(7) is weakly continuous at ,(8) is quasi continuous on ,(9) is quasi continuous at ,(10) is a.c.S on ,(11) is a.c.S at ,(12) is a.c.H on ,(13) is a.c.H at .

Proof. The equivalence of , , , and is proved in [4, 6]. With a bit more observation, we can prove that to are all equivalent. First we notice that To prove (28), we let . Then is a triangle triplet. So by Theorem 18, is a triangle triplet. Therefore, Thus, , as asserted. Now we will prove that , , , , and are equivalent.
Assume that is continuous at . Let . Then there exists a such that Now if and , then by (28) and (30), we obtain This shows that is uniformly continuous on .
It is easy to see that implies and implies .
We assume that holds. Let be the sequence in such that is strictly decreasing and converges to , and converges to . Therefore, if is given, there exists such that This proves . Since and are equivalent, we see that , , , , and are equivalent, as asserted.
It is true in general that every continuous function is quasi continuous. So it is easy to see that implies and implies . Next assume that holds. To show , let be given. Let . Then and are open set in containing and , respectively. Since is quasi continuous at , there exists a nonempty open set such that . Now we can choose so that and . This gives . Since and are equivalent, we obtain that , , , and are equivalent. Similarly, it is easy to see that implies , implies , implies , and implies . Since and are equivalent, it now suffices to show that each of and implies . First assume that holds. Let and let . Then is open in and contains . Since is a.c.S. at , there exists an open set containing such that Now we can choose so that and . Similarly if holds, then is a neighborhood of , so , and therefore we can choose so that and . This completes the proof.

The function in Example 22 shows that in the case of ultrametric-metric-preserving functions, the global continuity on and the local continuity at are not equivalent. In addition, the uniform continuity on and continuity on are not equivalent as can be seen from the function in Example 11. However, we still have the following result for the continuity at .

Theorem 26. Let be ultrametric-metric-preserving. Then the following statements are equivalent: (i) is continuous at ,(ii) is weakly continuous at ,(iii)for every , there exists an such that ,(iv) is quasi continuous at ,(v) is a.c.S. at ,(vi) is a.c.H. at .

Proof. We have that (i) implies (ii) is true in general. By the same argument that implies in Theorem 25, we see that (ii) implies (iii). Next assume that (iii) holds. To show that is continuous at , let be given. Then by (iii), there exists such that . Let and let . Since and , we obtain by Corollary 8 and Theorem 19 that This gives (i). Therefore, (i), (ii), and (iii) are equivalent. Since (i) implies (iv), (v), and (vi), it suffices to show that each of (iv), (v), and (vi) implies (iii). Since , it is amenable and we can use the same argument of the proof of Theorem 25 to show that (iv) implies (iii) (the same as implies ), (v) implies (iii) (the same as implies ), and (vi) implies (iii) (the same as implies ). This completes the proof.

Corollary 27. Let . If is discontinuous at , then there exists an such that for all .

Proof. This follows from (i) and (iii) in Theorem 26.

Example 28. Let be given by

First we will show that by applying Theorem 19. So we let . If , then for every . In particular, . If , then . So . It is easy to see that is weakly continuous at but is not continuous at . In fact is weakly continuous at every and is not continuous at any . This shows that we cannot replace continuity at in Theorem 26 by continuity at any other point . Similarly, and is quasi continuous on but is not continuous at .

Conflict of Interests

The authors declare that they have no competing interests.

Acknowledgment

The first author received financial support from The Thailand Research Fund (research Grant no. TRG5680052). The author takes this opportunity to thank The Thailand Research Fund for the support.

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