path components of a mapping space as homotopy class

Let \((X,\mathcal{T}_X)\) and \((Y,\mathcal{T}_Y)\) be topological spaces, where \(X\) is locally compact and \(\varphi: A \rightarrow Y\) a continuous map for \(A \subseteq X\) Then for the relative mapping space \(\mathrm{Map}_A(X,Y)\) with the compact open topology, there exists a Bijection

\begin{align*} \varphi: \{[X,Y]_\varphi \} \mapsto& \pi_0(\mathrm{Map}_{\varphi}(X,Y)) \end{align*}

corollary of:

• path inside the mapping space as homotopy

as two maps are homotopic relative to \(\varphi\) if and only if there exists a path

\begin{align*} \gamma: [0,1] \rightarrow \mathrm{Map}_{\varphi}(X,Y) \end{align*}