Definition 1: A map f:X[right arrow]Y is called a pre [alpha] g* closed map if f(A) is [alpha] g* closed in Y whenever A is [alpha] g* closed in X.
Theorem 1: Every homeomorphism is a pre [alpha] g* closed map.
By this definition, [theta] is an open and closed map
Balachandran, Semi-generalized closed maps
and Generalized semi- closed maps
Let S be the semigroup of all closed maps
like f : X [right arrow] X, under composition of maps:
Balachandran: Semi-generalised closed maps
and generalised semi-closed maps