International Journal of Computer Applications (0975 – 8887) Volume 89 – No 18, March 2014 12 On Generalized d-Closed Sets H. Jude Immaculate Research Scholar Nirmala College for Women Coimbatore. I. Arockiarani Department of Mathematics Nirmala College for Women Coimbatore. ABSTRACT In this paper we present a new class of sets and functions namely gd-closed sets, gd-irresolute functions in the light of d-open sets in topological spaces. Further some of their characterizations are investigated with counter examples. Keywords gd-closed sets, gd-continuous functions, gd-irresolute functions. 1. INTRODUCTION Many different forms of continuous functions have been introduced over years. Most of them involve the concept of g-closed sets, -open sets, semiopen sets, sg-open sets, etc. In 1987, Bhattacharya and Lahiri [5] introduced the class of semi-generalized closed sets. In 1990, Arya and Nour[3]defined generalized semiclosed sets. The concept of generalized closed sets was first initiated by Levine in 1970[12]. The notion of b-open sets was defined by D. Andrijevic in 1996[2].In this paper a new class of sets called gd-closed sets has been introduced using the concept of d-closed sets by I.Arockiarani et al[10]. Further we study the basic properties of gd-closed sets. Using this new concept of sets we have introduced new class of functions called gd-continuous and gd-irresolute functions. Some of its basic properties and composition of functions is also discussed here. Preliminaries: We present here relevant preliminaries required for the progress of this paper Definition 1.1: A subset A of a topological space (X,  ) is called 1. Preclosed set[15] if cl(int(A))  A, preopen set if A  int(cl(A)) 2.  -open set [17] if A  int(cl(int(A))),  -closed set if cl(int(cl(A)))  A. 3. Regular open set[19] if A=int(cl(A)), regular closed set if A=cl(int(A))) 4. Semiopen set [11] if A  cl(int(A)), semiclosed set if int(cl(A)  A. 5. Semi-pre-open set[1] if A  cl(int(cl(A))), semi-pre-closed set if int(cl(int(A)))  A. 6. d-open set[10] if A  scl(int(A))  sint(cl(A)) Definition 1.2: A subset A of a topological space (X,  ) is called 1. A generalized closed set[12] (briefly g-closed) set if cl(A)  U whenever A  U and U is open. 2. A generalized semiclosed set[3] (briefly gs-closed if cl(A)  U whenever A  U and U is open. 3. A -generalized closed set [13](briefly  g-closed) if  cl(A)  U whenever A  U and U is open. 4. A generalized preclosed set [14](briefly gp-closed) if pcl(A)  U whenever A  U and U is open. 5. A generalized pre regular closed set[9] (briefly gpr-closed) if pcl(A)  U whenever A  U and U is regular open 6. A -regular generalized closed set[12] (briefly  gr –closed) if  cl(A)  U whenever A  U and U is regular open 7. A regular generalized closed set [18](briefly rg-closed) if cl(A)  U whenever A  U and U is regular-open. 2. GENERALIZED D-CLOSED SETS Definition 2.1: A subset A of a space X is called Generalized d-closed set if dcl(A)  U whenever A  U and U is open. The class of all generalized d-closed sets is denoted by GDC(X). Definition 2.2: i. The gd-closure of a subset A of X is denoted by gdcl(A) is the smallest gd-closed set containing A. ii. The gd-interior of a subset A of X is denoted by gdint(A) is the largest gd-open set contained in A. Proposition 2.3[9]: The intersection of a open set and d-open set is d-open set. Proposition 2.4: i. Every closed set is gd-closed set. ii. Every d-closed set is gd-closed set.