relative mapping set

Let \((X,\mathcal{T}_X)\) and \((Y,\mathcal{T}_Y)\) be topological spaces, \(A \subseteq X\) the subspace topology and \(\varphi: A \rightarrow Y\) a continuous map.

Then the mapping space relative to \(\varphi\) is defined as subspace

\begin{align*} \mathrm{Map}_\varphi(X,Y) = \{f \in \mathrm{Map}(X,Y), f_{\vert A} = \varphi\} \end{align*}