stylistic:tone
Homotopy equivalence

Given two topological spaces X and Y, a homotopy equivalence between X and Y is a pair of continuous maps f : X → Y and g : Y → X, such that g ∘ f is homotopic to the identity map id and f ∘ g is homotopic to id. If such a pair exists, then X and Y are said to be homotopy equivalent, or of the same homotopy type. 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.