On Dislocated Metric Topology

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.


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][3][4][5][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 X be a set.

 
: d X X    0 , is called a distance function.Consider the following conditions, for all , , x y z  X , (d 1 ) ; It is obvious that any partial metric is a d-metric.
is called a d-membership relation (on X ) if it satisfies the following property for all x X  and , A B X  : xRA and A B  implies xRB .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 X be a nonempty set.Suppose that is a d-membership relation on R X and x u   is a collection of subsets of X for each x X  .We call   , x u R a d-neighbourhood system (d-nbhood system) for x if it satisfies the following conditions: such that for all yRV we have is called an HS-d-topological space where .
  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.

Redefinition of Definition 1.5.
In Proposition 1.1, it is proved that  , ,  X u R is an HS d-topological neighbourhood space.We remark that Property (Niii) can be replaced by the following condition: . One can easily verifies that X u R 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 X be a nonempty set.Suppose that is a d-membership relation on R X and x u   be a collection of subsets of X for each x X  .We call a d*-neighbourhood system (d*-nbhood system) for x if it satisfies the following conditions:

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 X be a nonempty set.Suppose that is a d-membership relation and for each x X  .We call x  an xd -topology on X iff it satisfies the following conditions: (dτ x 1) ; x X   (dτ x 2) , ; A is a be functions defined as follows: (dF x 1) x d -closed sets satisfies the conditions (dF x 1)-(dF x 3).

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.
x d -interior of a subset A of X is denoted and defined by:     : and Proof.The desired result is obtained from the following: x , then is defined.
(I) (dτ x 1) such that the conditions B(i), B(iii) and B(iv) are satisfied then ).

 
, , Definition 4.2.Let From Theorems 4.1 and 4.3, we obtain the following theorem without proof.
Theorem 4.4.Let xd X R  be an -topological space.
, then for each

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.(

1 .
The concepts of d*-topological neighborhood space and d-topological space are the same.be the family of all d*topological neighbourhood systems on X and let dT X be the family of all d-topologies on X .The proof is complete if we point out a bijection between of a subset A of X is denoted and defined by:

-topolo- gical neighborhood space. Now
, we state the following theorem without proof. where