Another Class of Distances and Continuous Quasi-Distances in Product Spaces
We construct a class of continuous quasi-distances in a product of metric spaces and show that, generally, when the parameter (as shown in the paper) is positive, is a distance and when , is only a continuous quasi-distance, but not a distance. It is remarkable that the same result in relation to the sign of was found for two other classes of continuous quasi-distances (see Peppo (2010a, 2010b) and Peppo (2011)). This conclusion is due to the fact that is a product space. For the purposes of our main result, a notion of density in metric spaces is introduced.
In this paper a quasi-distance on a set is defined as a function with the usual properties of a metric and a weaker version of the triangle inequality: where .
This function is not always continuous with respect to the -topology generated by itself in the same manner as by a distance. This is because the “open” balls , which form a base for a complete system of neighbourhoods of , are not always open sets in the -topology. Examples of “open” balls that are not open sets can be found in .
It is known that for every quasi-distance , there exists another quasi-distance , whose topology and uniformity are the same as those of and the open -balls are open sets .
But if any condition or relation is satisfied with respect to a quasi-distance , we do not know if, in general, the same condition or relation will be satisfied with respect to the “new” quasi-distance , even if is equivalent to .
For this reason, the continuity of a quasi-distance cannot be omitted without “danger.”
In fixed-point theory, some authors require the continuity of the used quasi-distance as complementary condition (see, among others, ).
The most interesting fact about the quasi-metric spaces is that, in many applications, they constitute a more general setting than the metric spaces without losing the good properties of these last spaces (see, among others, ).
In this paper we revisit this problem for another family of functions in a product space defined by with and we find the same result.
This makes us think that the conclusions reached are not coincidental.
We have reason to believe that a similar result holds for other classes of quasi-distances in product spaces.
Additionally, we notice that the iterated quasi-triangle constants of our quasi-distances, which we are calling generalized improved quasi-triangle inequality, improve the general one: (the two last coefficients are ; it is not a mistake).
In Section 2 we introduce the notion of a density condition in metric spaces for the purposes of our main result.
In Section 4 we give a counterexample proving the importance of the -density condition in part III of our main result.
2. Density in Metric Spaces
In the third part of our main result (Section 3) we will need metric spaces satisfying an additional property of density that we are calling -density condition.
Definition 1. For an and a positive number , the pair of metric spaces and satisfies the -density condition if the space contains at least three distinct points, , satisfying the relations and for every there exist at least three distinct points , satisfying the relations
In terms of segments, Definition 1 may be expressed as follows.
For an and a positive number , the pair of metric spaces and satisfies the -density condition if contains at least a segment, , containing a point dividing it at the ratio and, for every , contains at least a segment , time “longer” in than in and containing a point dividing it at the ratio .
By segment we mean the set defined by the following:
Definition 2. For two points in a metric space , the segment is the set of points satisfying
It is obvious that the extremities and belong to the segment , but it may only be reduced at its two extremities.
We note also that, despite appearances, if , , and , this does not mean, in general, that , as in the example below (of course, in the particular case when , we have ).
Example 3. Let be two strictly positive numbers satisfying , , a number satisfying , and a set containing whatever four points , , , and of . Define the function by
The two first properties of a distance are obviously satisfied.
So, to prove that is a distance, it suffices to prove the triangle inequality in all possible “triangles” formed with the points of , that is, , , , and (Figure 1).
In the “triangles” and , the “sides” are and the greatest one is , so the “triangle” inequality is satisfied.
In the “triangles” and , the “sides” are, respectively, and and the greatest one is, by our choice, ; or , and , so the “triangle” inequality is satisfied.
The points and belong to , because and . On the other hand , but, generally, .
We note that the -density condition, for an and a positive number , as it is required in the third part of our main result (Section 3), is not a very restrictive condition. For example, if is the usual distance in , , and , in order that the pair (; ) satisfies the -density condition, it suffices that contains the set , while contains the set as in Figure 2.
Clearly, every pair of normed vector spaces over satisfies the -density condition for each and for each positive number .
3. Main Result
Theorem 4. Let for be metric spaces, and real numbers, and positive numbers satisfying , and and two arbitrary points of . Then(i)the function defined by is a continuous quasi-distance on satisfying the generalized improved triangle inequality for every points;(ii)if or are all nonnegative or all nonpositive, is a distance;(iii)if and there exist two indexes with , and an such that the spaces and satisfy the -density condition, the continuous quasi-distance is not a distance.
Proof. (i) First, we will prove that is a quasi-distance and then that is a continuous function of its two variables with respect to the -topology.
It is clear that , that is symmetric, and that .
Let , , and .
Clearly, for whatever points and , we have
As is a distance, we can write, for whatever points , , ,
This means that is a quasi-distance with constant , and the same constant holds for whatever points. Let us show now that is continuous with respect to the -topology.
Inequalities (11) show that and are topologically equivalent. Hence to prove that is continuous, we will show that for every two sequences, and , converging, respectively, to and for , converges to .
For the points and , in the following, we will denote , , and .
If , an integer exists such that for , we have () (because the sequence converges to ).
For , , we have () and, consequently, converges to and ().
If , from we deduce That is, .
It follows that and converge to the same limit . We can conclude that , formed by the terms of the two sequences above, also converges to . Thus, is continuous and the “open” balls really are, in fact, open sets for the -topology.
(ii) If s are all nonnegative, for every and ; if s are all nonpositive, for every and (recall that if , ). As and are distances, is a distance too.
If , for every , so for every and ; that is, is a distance.
Suppose now that ; for two arbitrary points and , denoting again we have
So, for it follows that
This means that if , , and since the maximum of two distances is a distance, is a distance.
Notice that in this part of the theorem we did not need any supplementary density condition on the spaces .
(iii) Suppose now that and that there exist an and two indexes such that , , and the pair satisfies the -density condition.
We choose a satisfying the relation Such a exists because, passing into limit in (16) as , we obtain that is clearly true.
As satisfies the -density condition, there exist three points , satisfying the relations and , and three points , satisfying the relations
With the points and we construct the points , , and of (except the th and the th coordinates, all the other coordinates are equal in , and ) and will prove that
For the points and we have from (18) that so .
For the points and , because and , so .
For the points and , because and , so .
We are ready to prove now the inequality , transforming it equivalently to From and , we have , so is positive and we can equivalently square (because and ), (we square) Dividing by we have The last inequality is true from (16). The proof is over. So, is not a distance.
Example 5. For and , the function defined by is a continuous quasi-distance, but not a distance. Here, , , , , , and , while the function defined by is a distance.
Here, , , , , , , and .
4. A Counterexample
Showing the importance of the -density condition in part III of the theorem.
Let , be two metric spaces with distances, respectively, , , , , real numbers, and positive numbers satisfying, for , , , , , and .
For whatever , it is obvious that the pair does not satisfy the -density condition.
We will show that the function defined for every two points by is a distance.
For every three points of , we will show that the greatest “distance” (we put quotes because we do not yet know if is really a distance) between these points is inferior or equal to the sum of the two others.
4.1. First Case
The six “distances” are
As , we have , so
The greatest of the six “distances” is .
In each of the four possible “triangles,” , , , and , there is a side equal to , another is equal to , and the third is .
Therefore we have to prove only one “triangle inequality,” valid for the four triangles: or equivalently
But from we have and from we have , so and the triangle inequality is satisfied.
4.2. Second Case
The six “distances” are
As , we have , so , and .
The greatest of the six “distances” is . In each of the four possible “triangles,” , , , and , there is a side equal to , another is equal to , and the third is . Therefore we have to prove only one “triangle inequality,” valid for the four triangles: that we square equivalently
But from we have and from we have , so and the triangle inequality is satisfied. So, is a distance.
We note that if are normed vector spaces, , very slight modifications transform our main result in its normed version and that, in this case, no density condition is needed.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author would like to thank Redon A. Cabej for his valuable suggestions which improved the final version of this paper.
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