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:
(dτx1) 
(dτx2) 
(dτx3) 
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:
(dFx1)
;
(dFx2)
;
(dFx3) 
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 (dFx1)- (dFx3).
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) (dτx1) 
since
;
(dτx2) 
and
;
(dτx3) 
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
.
NOTES