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Introduction

The role of distance in understanding the world is undeniable. Our intuitive understanding of the concept of distance in the real world, however, is different from the one proposed in mathematics. Some of the properties that belong to our understanding of distance from the real world, such as symmetry and single-valuedness, are not necessarily established within certain abstract distances.

This will, in fact, be our main motivation for presenting a generalized concept of distance as a set-valued function in this paper. The notion of 2-metric spaces, as a possible generalization of metric spaces, was introduced by Gähler (Gähler, 1963), (Gähler, 1964), (Gähler, 1966). See also (Diminnie et al, 2017), (Ha et al, 1990) for further developments. The 2-metric d(x, y, z) is a function of 3 variables, and Gähler geometrically interpreted it as an area of a triangle with the vertices at x , y and z, respectively.

This led B. C. Dhage, in his PhD thesis in 1992, to introduce the notion of D-metric (Dhage et al, 2000) that does, in fact, generalize metric spaces. Subsequently, Dhage published a series of papers attempting to develop topological structures in such spaces and prove several fix point results.

Most of the claims, however, concerning the fundamental topological properties of D-metric spaces, are incorrect. In 2003, Mustafa and Sims demonstrated that in a strong remark (Mustafa & Sims, 2003). This led them to introducing the notion of a G-metric space (Mustafa & Sims, 2006), as a generalization of metric spaces. In this type of spaces, a non-negative real number is assigned to every triplet of elements.

In an attempt to generalize the notion of a G-metric space to more than three variables, Khan first introduced the notion of a K-metric, and later the notion of a generalized n-metric space (for any n ≥ 2) (Khan, 2012), (Khan, 2014). He also proved a common fixed point theorem for such spaces.

G-metric spaces were generalized to universal metrics in (Dehghan Nezhad & Mazaheri, 2010), (Dehghan Nezhad & Aral, 2011), (Dehghan Nezhad & Khajuee, 2013), (Dehghan Nezhad et al, 2017). The interpretation of the perimeter of a triangle is applied, but this time on G-metric spaces. Since then, many authors have obtained fixed point results for G-metric spaces.

The main purpose of this paper is a generalization of universal metric spaces into universal hypermetric spaces of the n-dimension (see (Kelly, 1975) for a discussion on hypermetric spaces). In the first part, we generalize an n-dimensional (n ≥ 2) hypermetric distance over an arbitrary non-empty set X. This hypermetric measures how separated all n points of the space are. The hyperdistance function is defined in any way we like, the only constraint being the simultaneous satisfaction of the three properties, viz non-negativity and positive-definiteness, symmetry and triangle inequality. In the second part, we discuss the concept of completness, with respect to this hypermetric, and the fixed point theorem, which play an important role in applied mathematics in a variety of fields. Examples show a fundamental difference between our results and the well-known ones. This concept is the first view of novel methods for selecting the clusters by hypermetric. The purpose definition is applicable for engineering science (for example, the theory of clustering).

By a strict order relation of a set X, we mean a binary relation ” < ”, which is transitive (α < β and β < γ implies α < γ), such that α < β and β < α cannot both hold. It is a strict total order relation, if for every α,β belonging to X, exactly one and only one of α < β, β < α or α = β holds. A group G is called left-ordered, if endowed with a strict total relation ” < ” which is left invariant, meaning that α < β implies γ + α < γ + β, for all α,β,γ ∈ G. We will say that G is bi-ordered, if it admits the left and right invariant properties simultaneously (historically, this has been called simple-ordered). We refer to the ordered pair (G, <) as an ordered group (Cohen & Goffman, 1949). From now on, we assume that 1 denotes the identity element of G. It should be noted that, for abelian additive groups, the identity element may be denoted by 0. This is common to an ordered group with the symbol ” ≤ ” that has the obvious meaning : α ≤ β means α < β or α = β. We denote G+ a set of non-negative elements of G, namely . Two positive elements, x, y, of an ordered group are relatively Archimedean if there are positive integers m, n such that mx ≥ y and ny ≥ x. If every two positive elements of an ordered group are relatively Archimedean, then the ordered group is Archimedean.

Every Archimedean ordered group is isomorphic to an ordered subgroup of the additive group of the real numbers. An ordered group G is order complete if every non-empty subset of G that has an upper bound has a least upper bound.

Universal hypermetric spaces of the dimension n

The goal of this section is to describe a few properties of the universal hypermetric spaces.

Definition 1. Let G be an ordered group. An ordered group metric (or OG-metric ) on a non-empty set X is a symmetric non-negative function d_G from X × X into G such that d_G(x, y) = 0 if and only if x = y and such that the triangle inequality is satisfied; the pair (X, d_G) is an ordered group metric space (or OG-metric space).

Now we first recall and introduce some notation. For n ≥ 2, let Xn denote the n-times Cartesian product and G be an ordered group. Let P*(G) denote the family of all non-empty subsets of G. We begin with the following definition.

Definition 2. Let X be a non-empty set. Let : Xⁿ → P*(G⁺) be a function that satisfies the following conditions:

Let A_i be subsets of X, i = 1, . . . , n. We define

We will use the following abbreviated notation: The function is called a universal ordered hypermetric group of the dimension n, or more specifically an UO_n-hypermetric (or U_n-hypermetric) on X, and the pair (X, ) is called an U_n-hypermetric space. For example, we can set G⁺ = or , where := ∪ {0} = {0, 1, 2, . . . } and := [0, +∞).

In the sequel, for simplicity we assume that G+ = . The following useful properties of a Un-hypermetric are easily derived from the axioms.

Proposition 1. (example) Let X = {a₁, . . . } be an -element set and = {1, . . . , }. Define

with,

and also assume A + B = A ∪ B, for all A, B ⊆ P(). Then is a U₂-hypermetric space.

Proof. It is sufficient to show that satisfies all the properties [(U1)], [(U2)], . . . , [(U5)] . The proofs of [(U1)], . . . , [(U4)], follow immediately from the definition of . We only need to show that satisfies the following relation

so we prove that in the following cases.

Proposition 2. Let (X, ) be a U_n-hypermetric space, then for any x₁, ..., x_n, a ∈ X it follows that:

Proposition 3. Let (X, ) be a Un-hypermetric space, then {0} ⊆ (x₁, ..., x_n) for all x₁, ..., x_n ∈ X.

Proof. By the condition (U4) of the definition of a U_n-hypermetric space, we have {0} = (x₁, ..., x_n) ⊆ (x₁, ..., x_n).

Proposition 4. Every Un-hypermetric space (X, ) defines a U₂- hypermetric space (X, ) as follows:

Proposition 5. Let e be an arbitrary positive real value number, and (X, d) be a metric space. We define an induced hypermetric,

Then (X, ) is a U₂-hypermetric space

Main results

Let (X, ) be a U_n-hypermetric space and be a partition of X. For each point p ∈ X, we denote a point in containing p, and we denote the equivalent relation induced by the relation by ∼

Definition 3. Let (X, ) be a U_n-hypermetric space. Let p₁, . . . , p_n ∈ X, and consider . A quotient U_n-hypermetric of the points of induced by is the function

given by .

Proposition 6. The quotient U_n-hypermetric induced by is well-defined and is a U_n-hypermetric on .

Proof. satisfies all the properties (U1) − (U4).

Let (X, ) be a U_n-hypermetric space of a dimension n > 2. For any arbitrary a in X, define the function on X_n−1 by

Then we have the following result.

Proposition 7. The function defines a U_n−1-hypermetric on X.

Proof. We will verify that satisfies the five properties of a U_n−1- hypermetric.

Proposition 8. Let f : X → Y be an injection from a set X to a set Y . If : Xⁿ → P* is a U_n-hypermetric on the set Y , then : Xⁿ → P*, given by the formula (x₁, . . . , x_n) = (f(x₁), . . . , f(x_n)) for all x₁, . . . , x_n ∈ X, is a U_n-hypermetric on the set X.

Proposition 9. Let (X, ) be any U_n-hypermetric space. Let λ be any positive real number. Then (X, ) is also a U_n-hypermetric space where

So, on the same X many U_n-hypermetrics can be defined, as a result of the procedure in which the same set X is endowed with different metric structures. Another structure in the next proposition is useful for scaling the U_n-hypermetric, so we need the following explanation.

For any non-empty subset A of , and λ ∈ R⁺ we define a set λ.A to be .

Proposition 10. Let (X, ) be any U_n-hypermetric space. Let Λ be any positive real number. We define (x₁, . . . , x_n) = λ.(x₁, . . . , x_n). Then (X, ) is also a U_n-hypermetric space.

A sequence {x_m} in a U_n-hypermetric space (X, ) is said to converge to a point s in X, if for any > 0 there exists a natural number N such that for every m₁, . . . , m_n−1 ≥ N,

then we write,

We say that a sequence {x_m} has a cluster point x if there exists a subsequence {x_mk } of {x_m} that converges to x.

Proposition 11. Let (X, ) and (X" , ) be two U_n-hypermetric spaces. Then a function f : X → X" is U_n-continuous at a point x ∈ X, if and only if it is U_n-sequentially continuous at x; that is, whenever sequence {x_m} is U_n-convergent to x one has {f(x_m)} which is U_m convergent to f(x).

Definition 4. Let (X, ) be a U_n-hypermetric space, and A ⊆ X. The set A is U_n-compact if for every U_n-sequence {x_m} in A, there exists a subsequence {x_mk } of {x_m} such that Un-converges to x₀ ∈ A.

Proposition 12. Let (X, ) and (X" , ) be two U_n-hypermetric spaces and f : X → X" a U_n-continuous function on X. If X is U_n-compact, then f(X) is U_n-compact.

Definition 5. Let (X, ) be a U_n-hypermetric space, then for x₀ ∈ X, r > 0, the U_n-hyperball with a center x₀ and a radius r is

Proposition 13. Let (X, ) be a U_n-hypermetric space, then for x₀ ∈ X, r > 0,

Proof. The proof of (i) is trivial. In (ii) it suffices to show that for every U_n hyperball B_Un (x, r) and every y ∈ B_Un (x, r)), there exists δ > 0 such that, y ∈ B_Un (y, δ) ⊆ B_Un (x, r). So let y ∈ B_Un (x, r). Then (x, . . . , x ; y) − Un(x, . . . , x ; x) ⊆ [0, r). Set

then δ > 0, and hence y ∈ B_U-n(y, δ).

Now let z ∈ B_Un (y, δ) , i.e, (y, . . . , y ; z) − (y, . . . , y ; y) ⊆ [0, δ), then

Thus, z ∈ B_Un (x, r), and hence B_Un (y, δ) ⊆ B_Un (x, r).

Proposition 14. The set of all -balls, = {B_Un (x, r) : x ∈ X, r > 0}, forms a basis for a topology (U_n) on X.

Definition 6. Let (X, ) be a U_n-hypermetric space. The sequence {x_n} ⊆ X is U_n-convergent to x if it U_n-converges to x in the U_n-hypermetric topology, (U_n).

Proposition 15. Let (X, ) be a U_n-hypermetric space. Then for a sequence {x_m} ⊆ X, and a point x ∈ X the following are equivalent:

Definition 7. Let (X, ), (Y, ) be universal hypermetric spaces of the dimensions n, m, respectively, a function f : X −→ Y is U_n,m-continuous at a point x₀ ∈ X, if f ⁻¹(B_Vm(f(x₀), r)) ∈ (U_n), for all r > 0.

We say f is U_n,m-continuous if it is U_n,m-continuous at all points of X; that is, continuous as a function from X with the (U_n)-topology to Y with the (V_m)-topology.

In the sequel, for simplicity we assume that n = m. Since U_n-hypermetric topologies are metric topologies, we have:

Definition 8. Let (X, ) and (Y, ) be two U_n-hypermetric spaces and f : (X, ) → (Y, ) be a function. The function f is called U_n-continuous at a point a ∈ X if and only if, for given > 0, there exists δ > 0 such that x₁, . . . , x_n−1 ∈ X and the subset relation (a, x₁, . . . , x_n−1) ⊆ [0, δ) implies that (f(a), f(x₁), . . . , f(x_n−1)) ⊆ [0, ).

A function f is U_n-continuous on X if and only if it is U_n-continuous at all a ∈ X.

Proposition 16. Let (X, ), (Y, ) be U_n-hypermetric spaces, a function f : X −→ Y is U_n-continuous at point x ∈ X if and only if it is Unsequentially continuous at x; that is, whenever {x_n} is U_n-convergent to x we have that (f(x_n)) is U_n-convergent to f(x).

Proposition 17. Let (X, ) be a U_n-hypermetric space. Then the function (z₁, z₂, ..., z_n) is jointly U_n-continuous in all n of its variables.

Definition 9. A map f : X −→ Y between U_n-hypermetric spaces (X, ) and (Y, is an iso-hypermetry when (x₁, ..., x_n) = (f(x₁), ..., f(x_n)) for all x₁, . . . , x_n ∈ X. If the iso-hypermetry is injective, we call it iso-hypermetric embedding. A bijective iso-hypermetry is called an isohypermetric isomorphism.

We discuss now about the concept of completeness of U_n-hypermetric spaces.

Definition 10. Let (X, ) be a U_n-hypermetric space, then a sequence {x_m} ⊆ X is said to be U_n-Cauchy if for every ε > 0, there exists N ∈ such that (x_m1 , x_m2 , ..., x_mn ) < ε for all m₁, m₂, ..., m_n ≥ N.

The next proposition follows directly from the definitions.

Proposition 18. In a U_n-hypermetric space, (X, ), the following are equivalent.

Corollary 1.

Definition 11. A U_n-hypermetric space (X, ) is said to be U_n-complete if every U_n-Cauchy sequence in (X, ) is U_n-convergent in (X, ).

Proposition 19. A U_n-hypermetric space (X, ) is U_n-complete if and only if (X, d_U ) is a complete metric space.

Definition 12. Let (X, ) and (Y, ) be two U_n-hypermetric spaces. A function f : X −→ Y is called a U_n-contraction if there exists a constant k ∈ [0, 1) such that (f(x₁), ..., f(x_n)) = k(x₁, ..., x_n) for all x₁, . . . , x_n ∈ X.

It follows that f is U_n-continuous because (x₁, ..., x_n) ⊆ [0, δ) with k $\ne$ 0 and δ := /k implies (f(x₁), ..., f(x_n)) ⊆ [0, ).

Theorem 1. Let (X, ) be a U_n-complete space and let T : X → X be a U_n-contraction map. Then T has a unique fixed point T(x) = x.

Proof. We consider x_m+1 = T(x_m), with x₀ being any point in X. We have, by repeated use of the rectangle inequality and the application of contraction property, the following:

for all m, s₁ ∈ which m < s₁ and k ∈ [0, 1).

Then we have

since

For m ≤ s₁ ≤ s₂ ∈ and (U5) implies that

now taking the limit as m, s₁, s₂ → +∞, we get

Now for m ≤ s₁ ≤ s₂ ≤ . . . ≤ s_n−1 ∈ , we will have

then {x_m} is a Cauchy sequence. By completeness of (X, ), there exists a ∈ X such that {x_n} is -convergent to a. The fact that the limit x_m is a fixed point of T follows the -continuity of T, and

Finally, if a and b are two fixed points, then

Since k < 1, we have (a, b, . . . , b) = {0}, so a = b and the fixed point is unique.

Conclusions

In this article, we have put forward a development of the results of U_n-hypermetric spaces, covering a variety of topics relevant for understanding their properties including completeness and the fixed-point theorem. We believe this work may be relevant from both the theoretical standpoint and the point of view of applications in contemporary problems such as those of clusterings which often appear in practice.

Acknowledgments

The author Nikola Mirkov is grateful for the financial support from the Ministry of Education and Science and Technological Development of the Republic of Serbia.

References

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Author notes

nmirkov@vin.bg.ac.rs

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