precomposition of homotopic maps

Let \((X, \mathcal{T}_X)\), \((Y, \mathcal{T}_Y)\) and \((Z, \mathcal{T}_Z)\) be topological spaces and \(f,f': X \rightarrow Y\) be homotopic relative to \(A \subseteq X\). Then for a continuous map \(g: Y \rightarrow Z\), it follows, that

\begin{align*} g \circ f, g \circ f': X \rightarrow Z \end{align*}

are homotopic relative to \(A\)

By assumption, there exists a homotopy \(H: [0,1] \times X \rightarrow Y\). composition with \(g\) results thus in a homotopy

\begin{align*} gH: [0,1] \times X \rightarrow Z \end{align*}

alternatively:

• path inside the mapping space as homotopy and postcomposition with \(g\)