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 \(\tau\) forms the function space \(C_\tau(Y,Z)\). For \(A\subset Y\), the set \(C(A,Z)\) of continuous functions from the space \(A\) to the space \(Z\) forms the underlying function space \(C_\zeta(A,Z)\) with the induced topology \(\zeta\). Topology \(\tau\) and the induced topology \(\zeta\) satisfies properties of splitting or admissibility and \(R_{A\subset Y}\)-splitting or \(R_{A\subset Y}\)-admissible properties respectively. In this paper we show that the underlying function space \(C_\zeta(A,Z)\) is topologically equivalent to the subspace \(C_{\varrho}(U_\circ), (V_\circ)\) of the function space \(C_\tau(Y,Z)\).