Homotopic maps

Remarks

The relation $\simeq$ is an equivalence relation on the set of continuous maps $X \to Y$. Reflexivity uses the constant homotopy $H(x, t) = f(x)$, symmetry reverses time, and transitivity runs two homotopies one after the other, each at double speed. The equivalence class of $f$ is its homotopy class $[f]$, and the set of classes is written $[X, Y]$. Homotopy is compatible with composition: if $f \simeq f'$ and $g \simeq g'$, then $g \circ f \simeq g' \circ f'$. Example: the maps $f(x) = x$ and $g(x) = 0$ from $[0, 1]$ to $\R$ are homotopic via $H(x, t) = (1 - t)x$. Non-example: the identity of the circle $S^1$ and a constant map $S^1 \to S^1$ are not homotopic, because they induce different maps on the fundamental group.