Chinese Journal of Mathematics
Volume 2015 (2015), Article ID 392190, 4 pages
http://dx.doi.org/10.1155/2015/392190

Research Article

On Quasimetrizability of Quasicone Metric Spaces

M. Aphane and S. P. Moshokoa

Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria 0008, South Africa

Received 11 May 2015; Accepted 23 June 2015

Academic Editor: Salvatore A. Marano

Copyright © 2015 M. Aphane and S. P. Moshokoa. 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 aim of this work is to extend interesting results on the metrizability of cone metric spaces as it appears in the literature. In this paper we appeal to quasiuniformities and uniformities to prove that a quasicone metric space is qausimetrizable, and from our results we will deduce that every cone metric space is metrizable; our approach is more on bitopological and topological properties and differs from the one used by the papers mentioned above but affirms some of their results.

1. Introduction

Ordered spaces and cones have a wide application in various branches of mathematics, for example, in applied mathematics, optimization theory, and computer science. There are many generalisations of metric spaces in the literature; for instance, we have pseudometric spaces and quasimetric spaces. One also obtains such a generalisation by replacing the codomain of a metric space when equipped with the usual metric with an ordered Banach space. Then one obtains a generalization of a metric space, referred to as a cone metric space; we refer to the paper by Long-Guang and Xian [1]. Since then many fixed points results in metric spaces were extended to the class of cone metric spaces; see [1], for example.

Furthermore, it is shown in the literature that cone metric spaces can be regarded as metric spaces topologically [2, 3] but that the study of fixed point results may still be worthwhile, since not all fixed point results in cone metric spaces can be reduced to similar results in metric spaces. The initial papers on cone metric spaces and quasicone metric spaces focused on fixed point results and recently it has become an interesting problem to find topological relationships between metric spaces and cone metric spaces. More specifically, the metrizability problem for cone metric spaces has received serious attention [2–5]. The purpose of the paper is to study bitopological properties of quasicone metric spaces, and we will show that a quasicone metric space can be regarded as a quasimetric space in the bitopological setup. However, it does not mean that all the results in quasicone metric spaces, especially those on fixed point theory, can be reduced to similar results in quasimetric spaces. We employ similar constructions as in the cone metric space but, more importantly, we also apply a totally new approach via quasiuniformities to explicitly construct a quasimetric space from a quasicone metric space. Since cone metric spaces are quasicone metric spaces and, not conversely, our results generalize and extend those in the literature. We shall show that all such constructions yield the same bitopological space; we finally discuss similar results in the setting of cone metric spaces and metric spaces.

2. Preliminaries

Let be a nonempty set; a function satisfying, for all ,

(i)(ii)

is called a quasipseudometric on and the pair is called a quasipseudometric space. If in addition to (i) and (ii) the function satisfies (iii) then is called pseudometric and the pair is referred to as a pseudometric space. It is well known that a pseudometric on is a metric if is equivalent to Now quasipseudometric that satisfies    if and only if is referred to as quasimetric on Clearly we see that every metric on is quasimetric on but not conversely. When is a quasimetric (metric) on , we shall refer to the pair as a quasimetric space (metric space).

Let be a nonempty set and let be a filter on We denote by . Suppose that the following hold for and on :

(a), for every .(b)For every there exists such that .

Then the filter is called a quasiuniformity on , and the pair is referred to as a quasiuniform space. Note that a quasiuniformity is a uniformity on if and only if, for every , the inverse of , denoted by , contains a member of So every uniformity on is a quasiuniformity, but the converse is not necessarily true.

It is well known [6] that given a quasiuniform space , we get a bitopological space. The bitopological space is obtained in the following manner. A subset (which is open) belongs to the topology induced by , if for every there exists such that , where . Let ; then is a quasiuniformity on , called the conjugate of Let the topology on induced by be denoted by ; then associated with a quasiuniform space we obtain the bitopological space

3. Main Results

The following definition is given in [4].

Definition 1. Let be an ordered vector space. An element is called an order unit if for each there exists a real number such that .

Definition 2 (see [4]). Let be a nonempty set and an ordered Banach space with a cone in such that and is closed. A function is a cone metric if

(i) if and only if ;(ii) for all ;(iii) for all .

In this case is called a cone metric space.

Example 3. Let be the set of real numbers, and . Put

For , we define

by

Then is a cone metric space.

Recently, the notion of a cone metric space has been extended to that of a quasicone metric space. Actually one such notion is found in [7] and later generalised by Shadda and Md Noorani in [8]. We recall the following.

Definition 4 (see [8]). Let be a nonempty set and let be an ordered vector space with cone . The function is called a quasicone metric if

(i) for all ;(ii) if and only if ;(iii) for all .

In this case is called a quasicone metric space.

Example 5. Let be the set of real numbers, and . Put

For , we define

by when and , otherwise. Then is a quasicone metric space.
Certainly a cone metric space is a quasicone metric space but not conversely.
We start by partially extending Theorem from [4], the proof is straight forward and therefore it is omitted.

Theorem 6. Let be a nonempty set, let be an ordered vector space with cone , and let . For a function , define by

(i)If is a quasicone metric, then is quasimetric.(ii)If is a cone metric, then is a metric.

Later on in the paper given a quasicone metric space by we denote the quasimetric on obtained in Theorem 6. For a vector space and , we define , where and Now given a quasicone metric space , define a function by , for all Then is a quasicone metric space. We shall refer to as a conjugate of Further, let the function be defined by , for all Then is a cone metric space. Given a cone metric space , we note that

Let us revisit Example 5.

Example 7. Let be the set of real numbers, and . Put

For , we define

by when and , otherwise. Then is a quasicone metric space. Note that is defined by

and certainly is a cone metric space.
We will write whenever Let be a quasicone metric space. For , let .

Proposition 8. Let be a quasicone metric space. Then the family forms a basis for the topology on

Given a quasicone metric space , let us denote the topology induced by with ; then the following is immediate.

Theorem 9. Let be a quasicone metric space; then we get a bitopological space

From Theorems 6 and 9, we easily deduce that a quasicone metric space is quasimetrizable.

Given a quasicone metric space and , define by

where if , or if and is a real number. Note that is quasimetric on whenever is a quasicone metric on ; moreover, is a metric on whenever is a cone metric on .

The following is an extension of Theorem   in [5].

Theorem 10. Let be a quasicone metric space and and a positive real number Then there exists a quasimetric which induces the same bitopological space; that is, one has , with and

The reader should note that the quasimetric constructed in Theorem 6 and the quasimetric constructed in Theorem 10 may differ when nontopological properties of the spaces involved are under consideration. For bitopological properties we have the following.

Theorem 11. Let be a quasicone metric space. Then the bitopological spaces , and are the same.

Next we provide a new result that proves that a quasicone metric space is quasimetrizable; we appeal to quasiuniformities.

The proof of the next result is similar to the proof of Lemma 13, so we omit the proof.

Theorem 12. Let be a quasicone metric space. Then there exists a quasiuniformity of such that .

Actually, we have the following.

Lemma 13. Let be a quasicone metric space and be fixed. Then is a countable base for a quasiuniformity on

Proof. Let and define . We will show that the collection is a base for a quasiuniformity on We first note that, for each , the set ; also, we have . Let and be positive integers such that . We will show that . Let . Find such that and . Clearly, ; hence Since , this shows that . It follows that is a countable base for the quasiuniformity on .

We recall the following regarding the quasimetrization problem of quasiuniform spaces.

Theorem 14. A quasiuniform space is quasimetrizable if and only if the quasiuniform space has a countable base.

Note that, from Lemma 13, if we denote the quasiuniformity , induced by with , then we have .

We now obtain the following.

Theorem 15. Let be a quasicone metric space and let be fixed. Then the bitopological space is quasimetrizable.

Proof. Let be a quasicone metric space, let be fixed, and let be the quasiuniformity induced on as obtained by Lemma 13. The quasiuniformity has a countable base; hence it is quasimetrizable by Theorem 14.
We now construct a quasimetric on associated with quasiuniformity in Lemma 13. The family in the proof of Lemma 13 is obtained from the quasicone metric space let Then

for all natural
We define a function by if and only if and if and only if for each Next we define by

over all finite sequences such that and The function is quasimetric on . Finally,

Corollary 16. Let be a cone metric space. Then the topological spaces , and are the same.

Corollary 17. Let be a cone metric space. Then there exists a uniformity of with a countable base such that .

The following confirms the already known result about cone metric spaces and the metrization problem for these spaces; see for example [3–5].

Theorem 18. Let be a cone metric space. Then the topological space is metrizable.

4. Contractions and Fixed Point Theory

Let be a quasicone metric space. A map is a contraction whenever

for all and . Note that if is a contraction, then

holds for all and , where is as in Theorem 6. Further, the following

holds for all and , where is the quasimetric on induced by defined earlier in the paper. So properties of quasicone metric spaces easily transfer to quasimetric spaces as discussed above in this section of the paper, so we can conclude that some fixed point results for contractive mappings in complete quasicone metric spaces [8, 9] may be reduced to similar ones in quasimetric spaces.

5. Conclusion

In the paper we show that every quasicone metric space can be regarded as a quasiuniform space, and we used this observation to construct a quasimetric space associated with the quasiuniform space, and this yielded the same bitopological space as the one induced by the quasicone metric space. It is instructive to mention that bitopological properties of quasimetric spaces, like pairwise normality and pairwise paracompactness also hold for quasicone metric spaces. We also showed that the quasicone metric space is quasimetrizable, and we explicitly constructed the quasimetric associated with the quasiuniformity. In the literature, for cone metric spaces different methods are employed and applied to construct a metric, such that the topologies induced by the cone metric and the resulting metric are the same. Similar constructions also work for quasicone metrics and quasimetric spaces and the resulting bitopological spaces are comparable; actually they are the same. In the paper we employed quasiuniformities and uniformities, to discuss a similar problem. Finally, we related our work to the ones done for cone metric spaces, and it is shown that our results extend, support, and confirm the work done for cone metric spaces.

Conflict of Interests

The authors declare they have no competing interests.

Authors’ Contribution

All the authors have contributed equally, read, and approved the submitted paper.

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