postcomposition of homotopic maps

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

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

are homotopic relative to \(f^{-1}[B]\)

By assumption, there exists a homotopy

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

thus by precomposition with \(( \mathrm{id}_{[0,1]} \times f\), we get a homotopy

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

alternatively:

• path components of a mapping space as homotopy class

Element \(\mathrm{Map}(Y, \mathrm{Map}([0,1], Z))\) and precomposition