Proposition

The path homotopy relation is preserved by composition with continuous maps. That is, if $f_0, f_1: I \rightarrow X$ are path-homotopic and $\phi:X \rightarrow Y$ is continuous, then $\phi \circ f_0$ and $\phi \circ f_1$ are path-homotopic.

Solution

Suppose $f_0$ and $f_1$ are paths from $x_0$ to $x_1$, and they are path-homotopic. Let $F: I \times I \rightarrow X$ be a path homotopy from $f_0$ to $f_1$. We claim that $\phi \circ F$ is a path homotopy from $\phi \circ f_0$ to $\phi \circ f_1$.

• $\phi \circ F$ is a composition of continuous functions, so it is continuous.
• $(\phi \circ F)(0, t) = \phi(f_0(t))$.
• $(\phi \circ F)(1, t) = \phi(f_1(t))$.
• $(\phi \circ F)(s, 0) = \phi(x_0)$.
• $(\phi \circ F)(s, 1) = \phi(x_1)$.

Therefore, $\phi \circ F$ is indeed a path homotopy from $\phi \circ f_0$ to $\phi \circ f_1$.