Deformation retraction/Definition

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Definition

A subspace, A, of a topological space (X,J) is called a deformation retract of X, if there exists a retraction[1][2], r:XA, with the additional property:

Recall that a retraction, r:XA is simply a continuous map where r|A=IdA (the restriction of r to A). This is equivalent to the requirement: riA=IdA.

Caution:Be sure to see the warnings on terminology

References


TODO: Mention something about how we must have a homotopy equivalence as a result. If riA=IdA then riA and IdX are trivially homotopic. As iArIdA we have the definition of a homotopy equivalence


  1. <cite_references_link_many_accessibility_label> 1.0 1.1 An Introduction to Algebraic Topology - Joseph J. Rotman
  2. <cite_references_link_many_accessibility_label> 2.0 2.1 Introduction to Topological Manifolds - John M. Lee