ABSTRACT

In this paper, we give a comment on the dislocated-neighbourhood systems due to Hitzler and Seda [1]. Also, we recover the open sets of the dislocated topology.

1. Introduction

In recent years, the role of topology is of fundamental importance in quantum particle physics and in logic programming semantics (see, e.g. [2-6]). Dislocated metrics were studied under the name of metric domains in the context of domain theory (see, [7]). Dislocated topologies were introduced and studied by Hitzler and Seda [1].

Now, we recall some definitions and a proposition due to Hitzler and Seda [1] as follows.

Definition 1.1. Let be a set. is called a distance function. Consider the following conditions, for all,

(d₁);

(d₂) if, then;

(d₃);

(d₄).

If satisfies conditions (d₁) - (d₄), then it is called a metric on. If it satisfies conditions (d₂) - (d₄), then it is called a dislocated metric (or simply d-metric) on.

Definition 1.2. Let be a set. A distance function is called a partial metric on if it satisfies (d₃) and the conditions:

(d₅) if and only if;

(d₆);

(d₇)for each.

It is obvious that any partial metric is a d-metric.

Definition 1.3. An (open) ball in a d-metric space with centre is a set of the form, where.

It is clear that may be empty in a d-metric space because the centre of the ball doesn’t belong to.

Definition 1.4. Let be set. A relation is called a d-membership relation(on) if it satisfies the following property for all and: and implies.

It is noted that the “d-membership”-relation is a generalization of the membership relation from the set theory.

In the sequel, any concept due to Hitzler and Seda will be denoted by “HS”.

Definition 1.5.Let be a nonempty set. Suppose that is a d-membership relation on and is a collection of subsets of for each. We call a d-neighbourhood system (d-nbhood system) for if it satisfies the following conditions:

(Ni) if, then;

(Nii) if, then;

(Niii) if, then there is a with such that for all we have;

(Niv) if and then.

Each is called an HS-d-neighborhood (HS d-nbhood) of. The ordered triple is called an HS-d-topological space where.

Proposition 1.1. Let be a d-metric space. Define the d-membership relation as the relation. For each, let be the collection of all subsets of such that. Then is an HS d-nbhood system for for each, i.e., is an HS d-topological neighbourhood space.

The present paper is organized as follows. In Section 2, we redefine the dislocated neighbourhood systems given due to Hitzler and Seda [1]. Section 3 is devoted to define the concept of dislocated topological space by open sets. In Section 4, we study topological properties of dislocated closure and dislocated interior operation of a set using the concept of open sets. Finally, in Section 5, we study some further properties of the well-known notions of dislocated continuous functions and dislocated convergence sequence via d-topologies.

2. Redefinition of Definition 1.5.

In Proposition 1.1, it is proved that is an HS d-topological neighbourhood space. We remark that Property (Niii) can be replaced by the following condition:

(Niii) * If, then for each.

One can easily verifies that satisfies (Niii) *.

According to the above comment, we introduce a redefinition of the concept of the dislocated-neighbourhood systems due to Hitzler and Seda [1] as follows.

Definition 2.1. Let be a nonempty set. Suppose that is a d-membership relation on and be a collection of subsets of for each. We call a d*-neighbourhood system (d*-nbhood system) for if it satisfies the following conditions:

(Ni) if, then;

(Nii) if, then;

(Niii)* if and, then;

(Niv) if and, then.

Each is called a d*-neighborhood of. If, then is called a d*-topological neighborhood space.

Now, we state the following theorem without proof.

Theorem 2.1. Let be a d-metric space. Define the d-membership relation as the relation iff there exists for which. Assume that and. Then is a d*-topological neighborhood space.

3. Dislocated-Topological Space

In what follows we define the concept of dislocatedtopological space (for short, d-topological space) by the open sets and prove that this concept and the concept of d*-topological neighborhood space are the same.

Definition 3.1. Let be a nonempty set. Suppose that is a d-membership relation and for each. We call an -topology on iff it satisfies the following conditions:

(dτ_x1)

(dτ_x2)

(dτ_x3) and.

Each is called a -open set. If is an -topology on for each, then is called a d-topology on. The triple is called an -topological space and the triple is called a d-topological space.

Definition 3.2. Let be an -topological space. is called a -closed iff is a - open..

Theorem 3.1. The concepts of d*-topological neighborhood space and d-topological space are the same.

Proof. Let be the family of all d*- topological neighbourhood systems on and let be the family of all d-topologies on. The proof is complete if we point out a bijection between and. Let and be functions defined as follows:, where for each and, where for each. One can easily verifies that these functions are well defined, and.

The following counterexample illustrates that the statement: iff may not be true.

Counterexample 3.1. Let and

.

Then is a d-membership relation. Since

, then, i.e. such that and.

We get the following theorem without proof.

Theorem 3.2. Let be a nonempty set. Suppose that is a d-membership relation and for each. Assume that satisfies the following conditions:

(dF_x1);

(dF_x2);

(dF_x3) and.

Then is a d-topology on, where. If is a dtopological space, then for each the family of all -closed sets satisfies the conditions (dF_x1)- (dF_x3).

4. Dislocated Closure and Dislocated Interior Operations

In the sequel we define the dislocated closure and dislocated interior operations of a set and study some topological properties of dislocated closure and dislocated interior operation.

Definition 4.1. Let be an -topological space. The -interior of a subset of is denoted and defined by:.

Remark 4.1. From Definition 4.1, if, then is undefined. If, then is defined.

Theorem 4.1. Let be an -topological space.

(A) If, then for each.

(B) If, then

(i);

(ii) for each;

(iii) for each;

(iv) or for each.

(v) if or.

Corollary 4.1. (1) If, then is a -open.

(2) If, then.

Theorem 4.2. If such that the conditions B(i), B(iii) and B(iv) are satisfied then

is an -topology on. The -membership relation is defined as iff.

Proof. The desired result is obtained from the following:

(I) (dτ_x1) since;

(dτ_x2) and

;

(dτ_x3) and, (from B(iii)-(iv)).

(II) and and (from I).

Definition 4.2. Let be an -topological space. The -closure of a subset of is denoted and defined by:.

If, then is undefined but if, then is defined.

Theorem 4.3. Let be an -topological space. Then for each,

.

Proof.

From Theorems 4.1 and 4.3, we obtain the following theorem without proof.

Theorem 4.4. Let be an -topological space.

(A) If, then for each.

(B) If, then

(i);

(ii) for each;

(iii);

(iv) or for each;

(v) if or.

Corollary 4.2. (1) If, then is a -closed.

(2) If, then.

5. Dislocated Continuous Functions and Dislocated Convergence Sequences via d-Topologies

Now, we define the dislocated continuous functions and dislocated convergence sequences. We also obtain a decomposition of dislocated continuous function and dislocated convergence sequences.

Definition 5.1. Let and be dislocated-metric spaces. A function is called d-continuous at iff such that. We say is d-continuous iff is d-continuous at each

Theorem 5.1. Let and be dislocated-metric spaces and be any function. Assume that (resp.) be the d-topological space obtained from (resp.). Then the following statements are equivalent:

(1) is d-continuous at.

(2)

(3) such that, where and are the d*-topological neighborhood systems obtained from and respectively.

(4) such that.

Proof. ((1)Þ(2)): Let. Then such that. Thus such that, i.e., , , then. Hence.

((2)Þ(1)): Let. Suppose that for each, such that. Now,. From the assumption, i.e., such that. Then. The contradiction demands that is d-continuous at.

(1) Û (4) and (2) Û (3) are immediate.

Definition 5.2. Let be a d-metric space. A sequence d-converges to if such that,.

Theorem 5.2. Let be a d-metric space and be the d-topological space obtained from it. Then the sequence d-converges to iff such that for each.

Proof. (Þ:) Let. Then there exists such that. From the assumption such that. Thus for each. So for each.

(Ü:) Let. Since, then. Thus such that for each ,i.e., for each. Hence.

REFERENCES

NOTES

^*Current address: Department of Mathematics, Faculty of Science, Northern Boarders University, Arar, KSA.