Proposition 1

Let X and Y be topological spaces. The relation “ $f$ is homotopic to $g$ ” is an equivalence relation in the set $\mathscr{C}(X; Y)$ of continuous mappings of X into Y.

Let $f$ be an element of $\mathscr{C}(X; Y)$. The mapping $f\circ$ pr$_1: X\times \mathbf{I}\rightarrow Y$ is a homotopy relating $f$ to $f$; this relation is therefore reflexive.

Let $f$ and $g$ be elements of $\mathscr{C}(X; Y)$ and $\sigma : X\times \mathbf{I}\rightarrow Y$ a homotopy joining $f$ to $g$. The mapping $\overline{\sigma}: X\times \mathbf{I}\rightarrow Y$ is then a homotopy joining $g$ to $f$; the relation considered is therefore symmetric.

Finally, let us prove that it is transitive. If $f,g$ and $h$ are elements of $\mathscr{C}(X; Y),\sigma$ a homotopy joining $f$ to $g$ and $\tau$ a homotopy joining $g$ to $h$, then $\sigma$ and $\tau$ are juxtaposed and $\sigma *\tau$ is a homotopy joining $f$ to $h$.

Let X and Y be topological spaces. The equivalence relation “$f$ is homotopic to $g$” in $\mathscr{C}(X; Y)$ (Prop. 1) is called the homotopy relation. The quotient set of $\mathscr{C}(X; Y)$ by this relation is denoted by [X; Y]. Its elements are called homotopy classes of continuous mappings of X into Y. The homotopy class of a continuous mapping $f: X\rightarrow Y$ will often be denoted by $[f]$.