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.