A subset A of a topological space X is said to be β-open  if A ⊂ Cl (Int (Cl (A))). A function f : X → Y is said to be β-irresolute  if for every β-open set V of Y, f-1(V) is β-open in X. In this paper we introduce weak and strong forms of β-irresolute functions and obtain several basic properties of such functions.
A new kind of sets called generalized μ-closed (briefly g μ-closed) sets are introduced and studied in a topological space by using the concept of generalized open sets introduced by Á. Császár. The class of all g μ-closed sets is strictly larger than the class of all μ-closed sets. Furthermore, g-closed sets (in the sense of N. Levine ) is a special type of g μ-closed sets in a topological space. Some of their properties are investigated. Finally, some characterizations of μg-regular and μg-normal spaces have been given.