Mehrdad Karavan jahromi

Let X be a Tychonoff space and (Y, d) be a metric space. Let C(X, Y ) be the space of continuous functions from X to Y . Given a function ϵ : X(0, ∞) and fC(X, Y ), define B(f, ϵ) = {gC(X, Y ) : d(f(x), g(x)) < ϵ(x)∀xX}. The fine topology τω has as a base all sets of the form B(f, ϵ), where ϵ runs over all elements from C+(X) We prove nontrivial generalizations of some known results concerning τΓ and τw on C(X). For example the following are equivalent 1. (C(X, Y ), τγ) = (C(X, Y ), τω); 2. X is a cb−space. Finally, Some topological properties of (C(X, Y ), τω) and (C(X, Y ), τγ) are studied too. Keywords: topology, fine topology, graph topology, pesudocompact, paracompact and cb−space