On Dislocated Metric Topology ()

Mohamed A. Ahmed, F. M. Zeyada, G. F. Hassan

Department of Mathematics, Faculty of Science, Assiut University, Qesm Than Asyut, Egypt.

**DOI: **10.4236/ijmnta.2013.24032
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Department of Mathematics, Faculty of Science, Assiut University, Qesm Than Asyut, Egypt.

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.

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M. Ahmed, F. Zeyada and G. Hassan, "On Dislocated Metric Topology," *International Journal of Modern Nonlinear Theory and Application*, Vol. 2 No. 4, 2013, pp. 228-231. doi: 10.4236/ijmnta.2013.24032.

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,

(d_{1});

(d_{2}) if, then;

(d_{3});

(d_{4}).

If satisfies conditions (d_{1}) - (d_{4}), then it is called a ** metric** on. If it **satisfies conditions** (d_{2}) - (d_{4}), 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 (d_{3}) and the conditions:

(d_{5}) if and only if;

(d_{6});

(d_{7})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_{x}1)

(dτ_{x}2)

(dτ_{x}3) 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_{x}1);

(dF_{x}2);

(dF_{x}3) 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_{x}1)- (dF_{x}3).

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_{x}1) since;

(dτ_{x}2) and

;

(dτ_{x}3) 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

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | P. Hitzler and A. K. Seda, “Dislocated Topologies,” Journal of Electrical Engineering, Vol. 51, No. 12, 2000, pp. 3-7. |

[2] | A. Batarekh and V. S. Subrahmanian, “Topological Model Set Deformations in Logic Programming,” Fundamenta Informaticae, Vol. 12, No. 3, 1998, pp. 357-400. |

[3] |
M. S. El Naschie, “A Review of E-Infinity Theory and the Mass Spectrum of High Energy Particle Physics,” Chaos, Solitons and Fractals, Vol. 19, No. 1, 2004, pp. 209-236. http://dx.doi.org/10.1016/S0960-0779(03)00278-9 |

[4] | M. S. El Naschie, “The Idealized Quantum Two-Slit Gedanken Experiment Revisted-Criticism and Reinterpretation,” Chaos, Solitons and Fractals, Vol. 27, No. 1, 2006, pp. 9-13. http://dx.doi.org/10.1016/j.chaos.2005.05.010 |

[5] | P. Hitzler, “Generalized Metrics and Topology in Logic Programming Semantics,” Ph.D. Thesis, National University of Ireland, University College, Cork, 2001. |

[6] | A. K. Seda, ‘Topology and the Semantics of Logic Programs,” Fundamenta Informaticae, Vol. 24, No. 4, 1995, pp. 359-386. |

[7] | S. G. Matthews, “Metric Domains for Completeness,” Ph.D. Thesis, University of Warwick, Warwick, 1986. |

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