Dislocated Fuzzy Metric Spaces And Associated Fuzzy Topologies Reny George 1 and M.S Khan 2 1 Present Affiliation Department of Mathematics Eritrea Institute of Technology, P.O Box 1056 Asmara, Eritrea, North East Africa 1 Permanent Affiliation : P.G Department of Mathematics and Computer Science St. Thomas College, Bhilai , Durg ( District ) Chhattisgarh State, India. 490006 Email Id. : renygeorge02@yahoo.com 2 Department of Mathematics and Computing P.O Box 36, Postal Code 123 Alkhod, Muscat, Sultanate of Oman. Email Id. : mohammad@squ.edu.om Abstract : The concept of dislocated fuzzy metric space is introduced and the associated fuzzy topologies are discussed. Generalized fuzzy versions of the Banach contraction mapping theorem is also proved. Keywords : Dislocated fuzzy metric space, dislocated neighbourhood, d-convergent. 1. Introduction: Zadeh’s introduction[15] of the notion of fuzzy sets laid down the foundation of fuzzy mathematics. In the last two decades there were a tremendous growth in fuzzy mathematics. Many fixed point theorems for contractions in fuzzy metric spaces appeared (see [1],[2],[4],[5],[8-10],[12-14] ). The role of topology in logic programming has come to be recognized in recent years. In particular topological methods are employed in order to obtain fixed point semantics for logic programs. Motivated by this fact Hitzler and Seda [6] introduced the concept of dislocated metric space and studied dislocated topologies associated with it. They also proved a generalized version of Banach contraction mapping theorem which was applied to obtain fixed point semantics for logic programs. In this paper we have introduced the concept of dislocated fuzzy metric spaces and studied the fuzzy topology associated with it. We have also proved two fixed point theorems in DFM-Space, which extends and generalizes the results of Grabiec [3], Gregory and Romuguera [4], Gregory and Sapena [5], Mihet [8] and Radu [10]. 2. Preliminaries Definition 2.1 [11] : A binary operation ∗ : [0,1] x [0,1] → [0,1] is a continuous t-norm if ([0,1], ∗ ) is an abelian monoid with unit one such that, for all a,b,c,d in [0,1], a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d. Definition 2.2 [6]: Let X be a set. A relation ) ( X P X × ⊆ f is called a d-membership relation on X if it satisfies the following property: x f A and A ⊂ B implies x f B ∀ x ∈ X, and A,B ∈ P(X), where P(X) is the power set of X . If x f A we read it as x is below A. Definition 2.3 [6]: Let X be a set, f be a d- membership relation on X. For each X x ∈ , let U x be the collection of all subsets of X satisfying the following conditions: (N 1 ) if U ∈ U x then x f U. (N 2 ) if U,V ∈ U x then U ∩ V ∈ U x . (N 3 ) if U ∈ U x then there is a V ⊂ U with V ∈ U x such that for all y f V we have U ∈ U y . (N 4 ) if U ∈ U x and U ⊂ V then V ∈ U x . Then (U x , f ) is called a d-neighborhood system for x and each U∈ U x is called a d-neighborhood of x. 20