Homeomorphism between the underlying function space and the subspace of the function space.

Abstract

The set of continuous functions from topological space Y to topological space Z endowed with topology T forms the function space C_T(Y;Z). For A ⊂Y , the set C(A;Z) of continuous functions from the space A to the space Z forms the underlying function space Cζ(A;Z) with the induced topology ζ. Topology T and the induced topology ζ satisfies properties of splitting or admissibility and R_A⊂Y -splitting or R _A⊂Y -admissible properties respectively. In this paper we show that the underlying function space Cζ(A;Z) is topologically equivalent to the subspace Cϱ(U∘ V∘) of the function space C_T(Y;Z).