Mathematica Moravica Vol. 14 - 1 (2010), 61–67 The γ -open Open Topology for Function Spaces S. Ganguly and Ritu Sen Abstract. In this paper we have introduced the notion of γ- open open topology and proved some properties which the topology does possess. We have also introduced the concept of convergence of nets in γH(X) (where γH(X) is the set of all self γ- homeomorphisms on a topological space X) and showed when γH(X) is complete. 1. Introduction A set-set topology is one which is defined as follows : Let (X, τ ) and (Y,τ ∗ ) be two topological spaces. Let U and V be collections of subsets of X and Y respectively. Let F ⊂ Y X be a collection of functions from X into Y . We define, for U ∈U and V ∈V , (U, V )= {f ∈ F : f (U ) ⊂ V }. Let S (U, V )= {(U, V ): U ∈U ,V ∈ V}. If S (U, V ) is a subbasis for a topology τ (U, V ) on F , then τ (U, V ) is called a set-set topology. The most commonly discussed set-set topologies are the compact-open topology, τ co , which was introduced in 1945 by R.Fox [4] and the point-open topology, τ p . For τ co , U is the collection of all compact subsets of X and V the collection of all open subsets of Y , while for τ p , U is the collection of all singletons in X and V the collection of all open subsets of Y . In section 2 of this paper, we shall introduce and discuss the γ open-open topology for function spaces. We shall also show which of the desirable properties τ γoo possesses. In section 3, we shall introduce the notion of convergence of nets in (γH (X ),τ γoo ) (where γH (X ) is the collection of all self γ -homeomorphisms on X ) and the completeness of γH (X ). Throughout this paper, (X, τ ) (simply X ) and (Y,τ ∗ ) always mean topo- logical spaces. Let S be a subset of X . The closure (resp. interior) of S will be denoted by cl(S ) (resp. int(S )). A subset S of X is called a semi-open set [7] if S ⊆ cl(int(S )). The complement of a semi-open set is called a semi-closed set. The family of all semi-open sets in a topological space (X, τ ) will be denoted by SO(X ).A subset M (x) of a space X is called a semi-neighborhood of a point x ∈ X 2000 Mathematics Subject Classification. Primary: 54C35. Key words and phrases. γ-open sets, γ-open open topology, γH(X)–the set of all self γ-homeomorphisms on X, γ- convergence, γ-regular. c 2010 Mathematica Moravica 61