Characterization of Function Spaces and the Underlying Spaces

Abstract

The set of continuous functions from topological space Y to topological space Z when endowed with a topology forms a function space. This function space inherits properties of topological spaces from space Z such as the lower separation axioms. Topologies defined on the function spaces are either splitting or admissible. A topology is splitting if the continuity of a map defined on the product of topological spaces X and Y to space Z, implies the continuity of a map defined on a space X to the function space. A topology is admissible if the continuity of a map defined on a space X to the function space, implies the continuity of a map defined on the product of spaces X and Y to another space Z. A property of topological space is hereditary if it is inherited by the subspace and hereditarily if it is found on the subspace only. Normality is a hereditarily property, the lower separation axioms are hereditary, while compactness is hereditary with respect to closed subsets. These properties have not been shown to be hereditary or hereditarily on the subspaces of function spaces. This research study defines the underlying set of continuous functions from set A subset of space Y, to space Z. It shows that the underlying set when endowed with a topology fonns the underlying function space. Topologies defined on the underlying set are shown to satisfy the condition of splitting and admissibility and have been named R ACY -splitting topology and R ACY -admissible topology respectively. The study also shows how properties of topological spaces, splitting and admissible topologies on the function space relate to those on the underlying fimction space. To achieve this, properties of continuous functions and set theory have been used. The results obtained are used to identify hereditary and hereditarily topological properties on function space and its subspaces.