Topologies on the underlying function space

For topological spaces Y and Z, C(Y,Z) represent the set of continuous functions from space Y to space Z. Let A be an arbitrary intersection of non-empty open subsets of Y. In this paper, we develop the set C(Y,Z) of continuous functions from space A to space Z and construct topologies on this set to form the underlying function space C_ζ(Y,Z) of the function space C_τ(Y,Z). We define continuous maps between the spaces X, A and C_ζ(Y,Z), and show that topologies defined on the set C(A,Z)are either R_A⋐Y-splitting or R_A⋐Y-admissible.