Given two topological spaces M and N, place an equivalence relationship on the continuous maps f:M->N using homotopies, and write f_1 ~f_2 if f_1 is homotopic to f_2. Roughly speaking, two maps are homotopic if one can be deformed into the other. This equivalence relation is transitive because these homotopy deformations can be composed (i.e., one can follow the other).

homotopy | homotopy group | homotopy theory | topological space