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

• Edward Njuguna Muturi Egerton University

### 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 $$C_\tau(Y,Z)$$.