Heredity of Lower Separation Axioms on Function Spaces

Abstract

The set of continuous functions from topological space Y to topological space Z endowed with a topology forms the function space. For A subset of Y , the set of continuous functions from the space A to the space Z forms the underlying function space with an induced topology. The function space has properties of topological space dependent on the properties of the space Z , such as the T0 , T1 , T2 and T3 separation axioms. In this paper, we show that the underlying function space inherits the T0 , T1 , T2 and T3 separation axioms from the function space, and that these separation axioms are hereditary on function spaces.