We introduce the different notions of a new class of continuous functions called generalized semi Lambda (gs ) continuous function in topological spaces. Its properties and characterization are also discussed. 1. Introduction In 1986, Maki [1] continued the work of Levine and Dunham on generalized closed sets and closure operators by introducing the notion of -sets in topological spaces. A -set is a set which is equal to its kernel (= saturated set), that is, to the intersection of all open supersets of . Arenas et al. [2] introduced and investigated the notion of -closed sets and -open sets by involving -sets. In 2008？Caldas et al. [3] introduced generalized closed sets ( g, -g, g ) and their properties. They also studied the concept of closed maps. In 2007, Caldas et al. [4] introduced the concept of irresolute maps. In this paper, we establish a new class of maps called gs continuous function and study its properties and characteristics. Throughout this paper, ( ), ( ), and ( ) (or simply , and will always denote topological spaces on which no separation axioms are assumed unless explicitly stated. , , , , gs , and gs denote the interior of , closure of , lambda interior of , lambda closure of , gs Lambda closure of and gs Lambda interior of , respectively. 2. Preliminary Definitions Definition 1. A subset of a space ( ) is called (1)a semiopen set [5] if , (2)a preopen set [6] if ,(3)a regular open set [6] if . The complement sets of semi open (resp., preopen and regular open) are called semi closed sets (resp., preclosed and regular closed). The semiclosure (resp., preclosure) of a subset of denoted by sCl( ), (pCl( )) is the intersection of all semi closed sets (pre closed sets) containing . A topological space ( ) is said to be (1)a generalized closed [7] if Cl( ) , whenever and is open in ( ),(2)a closed [8] if Cl( ) , whenever and is g-open in ( , ), (3)semigeneralized closed (denoted by sg-closed) [9] if sCl( , whenever and is semi open in ( ), (4)generalized semiclosed (denoted by gs-closed) [9] if sCl( , whenever and is open in ( ), (5)a subset of a space ( ) is called -closed [2] if , where is a -set and is a closed set, (6)a subset of ( ) is said to be a g closed set [3] if Cl( whenever , where is open in ( ), (7)a subset of ( ) is said to be a -g closed set [3] if whenever , where is open in ( ), (8)a subset of ( ) is said to be a g closed set [3] if whenever , where is open in ( ), (9)a subset of ( ) is said to be a gs closed set [10] if whenever , where is semi open in . The complement of the above closed sets are called its respective
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