Homotopic maps

Summary

Two continuous maps $f, g \colon X \to Y$ are homotopic, written $f \simeq g$, when there is a homotopy from $f$ to $g$. Being homotopic is an equivalence relation on the set of continuous maps from $X$ to $Y$. Its equivalence classes are the homotopy classes, and the set of them is written $[X, Y]$. Algebraic topology studies spaces through such sets of homotopy classes.