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Deformation Retraction

Definition: Let A be a subspace of X. We say that A is a deformation retract of X if the identity map of X is homotopic to a map that carries all of X into A, such that each point of A remains fixed during the homotopy. This means that there is a continuous map H: XxI > X such that H(x, 0) = x and H(x, 1) is in A for all x in X, and H(a, t) = a for all a in A. The homotopy H is called a deformation retraction of X onto A. The map r: X > A defined by the equation f(x)  H(x, 1_ is a retraction of x onto A, and H is a homotopy between the identity map of X and the map j(r(x)), where j: A > X is inclusion.
Source: Topology (second edition) by James R. Munkres
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