Homotopy equivalence

Homotopy equivalence

Given two topological spaces X and Y, a homotopy equivalence between X and Y is a pair of continuous mapsf : X → Y and g : Y → X, such that g ∘ f is homotopic to the identity map id_X and f ∘ g is homotopic to id_Y. If such a pair exists, then X and Y are said to be homotopy equivalent, or of the same homotopy type. This relation of homotopy equivalence is often denoted ${\displaystyle \simeq }$. Intuitively, two spaces X and Y are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations. Spaces that are homotopy equivalent to a point are called contractible.