Csaszar introduced the concept of generalized topological space and a new open set in a generalized topological space called -preopen in 2002 and 2005, respectively. Definitions of -preinterior and -preclosuer were given. Successively, several studies have appeared to give many generalizations for an open set. The object of our paper is to give a new type of generalization of an open set in a generalized topological space called -semi-p-open set. We present the definition of this set with its equivalent. We give definitions of -semi-p-interior and -semi-p-closure of a set and discuss their properties. Also the properties of -preinterior and -preclosuer are discussed. In addition, we give a new type of continuous function in a generalized topological space as -semi-p-continuous function and -semi-p-irresolute function. The relationship between them are showen. We prove that every -open ( -preopen) set is an -semi-p-open set, but not conversely. Every -semi-p-irresolute function is an -semi-p-continuous function, but not conversely. Also we show that the union of any family of -semi-p-open sets is an -semi-p-open set, but the intersection of two -semi-p-open sets need not to be an -semi-p-open set.
