d-Neighborhood system and generalized F-contraction in dislocated metric space

The following question was put forth in Hitzler Thesis.

(Question 1.1). Question: Is there a reasonable notion of d-open set corresponding
to the notions of d-neighborhood, d-convergence and d-continuity.

We provided an answer for the above open question by constructing below theorems.

Theorem 3.1

Letbe a d -topological space. Definefor each. Thenis a topology on X.

Proof

Clearly contains X and

Let be an indexed family of non-empty elements of .

Let which implies that for some

Thus there exists such that . Which implies that

Let be any finite intersection of elements of

We have to prove that To obtain this, first we prove that if then Let .

Which implies that and then there exists and there exists

Which implies and .

Thus Hence by induction, we get

Definition 3.2

Let be a d-topological space and be a d–open if for every there exists

Definition 3.3

Let be a d-topological space and is d–open then is d–closed.

Definition 3.4

Let be a d-topological space and A point x in A is called an interior point of A if

Remark

Interior point of A is an open set.

Definition 3.5

Let be a d-topological space and A point x in X is said to be limit point of A if for every there exist in A such that

Definition 3.6

Let (X, d) be a d-metric space and . If there is a number such that then f is called a contraction.

Definition 3.7

(
Sarma and Kumari 2012]) Let (X, d) be a d-metric space and be a mapping. Write and . We call points of Z(f) as coincidence point of f. Clearly every point of Z(f) is a fixed point of f but the converse is not necessarily true.

Theorem 3.8

A subsetis said to be d–closed iff a netin F d–converges to x then

Proof

Suppose is d-closed.

Let be a net in F such that lim

We shall prove that

Let us suppose which implies that which is open.

Thus there exists such that

As there exists such that .

Since lim there exists such that for .

Hence A contradiction.

It follows that

Conversely, assume that if a net in F d–converges to x then

We shall prove that is d-closed.

is d-open.

For this we have to prove that for every there exists such that

Suppose for some there exists such that

Let

As is a direct set under set inclusion

Thus is a net.

Let .

If

Thus implies that .

It follows that lim.

Which implies that A Contradiction.

So for all there exists such that

Which completes the proof.

Remark

For each is a d-neighborhood of x.

Theorem 3.9

Letbe a d -topological space and letbe the collection of all subsets U of X such thatThenis said to be a basis for a topology on X if

(i)
For eachthere existssuch that

(ii)
Ifthere existsand

Proof

(i) is clear.

Since implies

So there exists such that

Since balls are d-neighborhood, choose

Then and

Lemma 3.10

Let X be any set andbe basis for the topologiesandrespectively. Then the following are equivalent.

(i)
finer than

(ii)
and each basis elementwiththere exists a basis elementsuch thatand

Theorem 3.11

Letbe the topology induced from the d -topological spaceobtained from d -metric as in Proposition 2.4,be the topology induced by the d -metric then

Proof

Let Then the collection is a basis for and is a basis for Clearly since is a d-neighborhood.

Let and such that

Since there exists such that

Which implies .

So Hence

Theorem 3.12

Letbe a d -topological space andandthe following are equivalent, assumefor every

(1)
There existssuch that lim

(2)
For everythere existsin A such that

Proof

Let there exists such that

Since (1) holds, lim.

Which implies that, there exists N such that

Let and then .

It follows that

So Hence (2) holds.

Assume that (2) holds. Let there exists in A such that

i.e there exists such that

Let .

Which implies that and .

Hence .

Which yields lim Hence (1) holds.

Theorem 3.13

Letbe the d -topological space obtained from d -metricas in Proposition 2.4 .Then balls are d -open.

Proof

Let be a ball with center at x and radius .

It sufficies to prove that is d-open.

i.e we shall prove for every there exists such that

Since implies .

Choose .

As is a d-neighborhood, now let .

So it is sufficient to prove that .

Let .

This implies that .

Then .

It follows that .

Hence

Theorem 3.14

Letbe a d -topological space obtained from d -metricas in Proposition 2.4. Thenis a Haussdorff space.

Proof

Suppose .

Let us choose

Let be the d-neighborhoods of x and y respectively.

It sufficies to prove .

Let .

Which implies that and .

So and .

It follows .

Which is a contradiction.

Theorem 3.15

Letbe a d -topological space obtained form d -metricas in Proposition 2.4. Then singleton sets are d -closed in

Proof

Let , we have to prove that is d-closed or it is sufficies to prove is d-open.

i.e for each there exists such that

Since implies .

Which yields .

Thus, there is a d-neighborhood, such that .

Hence is d-closed.

Corollary 3.16

Letbe a d -topological space obtained form d -metric. Thenis a-space.

Corollary 3.17

Letbe a d -topological space. Then the collectionis an open base at x for X.