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Concept
Retraction (topology)
Summary
In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deformation retraction is a mapping that captures the idea of continuously shrinking a space into a subspace.
An absolute neighborhood retract (ANR) is a particularly well-behaved type of topological space. For example, every topological manifold is an ANR. Every ANR has the homotopy type of a very simple topological space, a CW complex.
Let X be a topological space and A a subspace of X. Then a continuous map
is a retraction if the restriction of r to A is the identity map on A; that is, for all a in A. Equivalently, denoting by
the inclusion, a retraction is a continuous map r such that
that is, the composition of r with the inclusion is the identity of A. Note that, by definition, a retraction maps X onto A. A subspace A is called a retract of X if such a retraction exists. For instance, any non-empty space retracts to a point in the obvious way (the constant map yields a retraction). If X is Hausdorff, then A must be a closed subset of X.
If is a retraction, then the composition ι∘r is an idempotent continuous map from X to X. Conversely, given any idempotent continuous map we obtain a retraction onto the image of s by restricting the codomain.
A continuous map
is a deformation retraction of a space X onto a subspace A if, for every x in X and a in A,
In other words, a deformation retraction is a homotopy between a retraction and the identity map on X. The subspace A is called a deformation retract of X. A deformation retraction is a special case of a homotopy equivalence.
A retract need not be a deformation retract. For instance, having a single point as a deformation retract of a space X would imply that X is path connected (and in fact that X is contractible).
Note: An equivalent definition of deformation retraction is the following.
Official source
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