Homotopy equivalence

A relation between topological spaces in which maps in both directions have composites homotopic to the respective identity maps. It identifies spaces with the same homotopy type, even when their shapes differ.

Homotopy: A continuous deformation of one map into another through a family of maps. The composites need only deform continuously to identity maps, not equal them.
Topological space: A set equipped with a topology that specifies its open subsets. Homotopy equivalence compares spaces through continuous maps.
Contractible space: A space homotopy equivalent to a one-point space. Its contraction provides a direct example of the maps and homotopies in the definition.
Homeomorphism: A continuous bijection between topological spaces whose inverse is continuous. Homotopy equivalence permits spaces that are not homeomorphic.
Homotopy inverse: A map whose composites with a given map are homotopic to the identity maps on both spaces. It supplies the reverse map required for a homotopy equivalence.
Continuous map: A function between topological spaces whose inverse images of open sets are open. Both directions of an equivalence must be continuous maps.
Deformation retract: A subspace onto which a space continuously deforms while fixing the subspace throughout. The inclusion and retraction give a standard way to prove homotopy equivalence.
Homology equivalence: A map inducing isomorphisms on homology groups in every degree. Homology equivalence is an algebraic test that need not establish homotopy equivalence.
Identity map: A function from a set to itself that sends every element to itself. The composites in the definition must be homotopic to each space’s identity map.
Product topology: The topology on a product of spaces generated by products of open sets. Homotopies can be expressed as continuous maps from a product with the interval.

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