In topology, a retraction, as the name suggests, "retracts" an entire space into a subspace. A deformation retraction is a map which captures the idea of continuously shrinking a space into a subspace.

Formal definitions

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Let X be a topological space and A a subspace of X. Then a continuous map $r:X\; \backslash to\; A$ is a retraction if the restriction of r to A is the identity map on A; that is, $r(a)=a$ for all a in A. Note that a retraction maps X onto A. In this case, A is called a retract of X.

A continuous map $d:X\; \backslash times1\backslash to\; X$ is a deformation retraction if, for every x in X, a in A, and t in 1,

$d(x,0)\; =\; x$

$d(x,1)\; \backslash in\; A$

$d(a,t)\; =\; a$.

A deformation retract is thus a homotopy between the identity map on X and a retraction of X onto A. A is called a deformation retract of X.

Note that although homotopy is an equivalence relation between maps, deformation retraction is not an equivalence relation between spaces. Generally one space is a proper subset of the other.

Any topological space which deformation retracts to a point is contractible. Contractibility, however, is a weaker condition, as contractible spaces exist which do not deformation retract to a point.

Topology

Деформационный ретракт