Homotopy Equivalence
Given two spaces X and Y, we say they are homotopy equivalent or of the same homotopy type if there exist continuous maps f : X → Y and g : Y → X such that g ∘ f is homotopic to the identity map idX and f ∘ g is homotopic to idY.
The maps f and g are called homotopy equivalences in this case. Every homeomorphism is a homotopy equivalence, but the converse is not true: for example, a solid disk is not homeomorphic to a single point, although the disk and the point are homotopy equivalent.
Two spaces X and Y are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations. For example, a solid disk or solid ball is homotopy equivalent to a point, and R2 − {(0,0)} is homotopy equivalent to the unit circle S1. Spaces that are homotopy equivalent to a point are called contractible.
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