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Definition: Let f: X -> Y and g: Y -> X be continuous maps. Suppose that the map (g o f): X -> X is homotopic to the identity map of X, and the map (f o g): Y -> Y is homotopic to the identity map of Y. Then the maps f and g are called homotopy equivalences, and each is said to be a homotopy inverse of the other.
Source: Topology (second edition) by James R. Munkres