Deformation retract

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

1 Retract
2 Neighborhood retract
3 Deformation retract
4 References

[edit] Retract

Let X be a topological space and A a subspace of X. Then a continuous map $r:X \to A$ is a retract 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, and r is called a retraction.

[edit] Neighborhood retract

Let X be a topological space and A a subspace of X. If there exists an open set U such that $A \subset U \subset X$ and A is a retract of U, then A is called a neighborhood retract of X.

[edit] Deformation retract

A continuous map $d:X \times [0, 1] \to X$ is a deformation retract if, for every x in X, a in A, and t in [0, 1],

d (x,0) = x

$d(x,1) \in A$

d (a, t) = a

A deformation retraction 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. However, deformation retraction is related to the notion of homotopy equivalence, in the sense that two spaces are homotopy equivalent if and only if they are both deformation retracts of a single space. This is an equivalence relation.

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.