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.
[1] 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.
Every ANR has the homotopy type of a very simple topological space, a CW complex.
For instance, any non-empty space retracts to a point in the obvious way (any constant map yields a retraction).
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 deformation retraction is a special case of a homotopy equivalence.
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.
In this formulation, a deformation retraction carries with it a homotopy between the identity map on X and itself.
In other words, a strong deformation retraction leaves points in A fixed throughout the homotopy.
(Some authors, such as Hatcher, take this as the definition of deformation retraction.)
consisting of closed line segments connecting the origin and the point
for n a positive integer, together with the closed line segment connecting the origin with
Now let A be the subspace of X consisting of the line segment connecting the origin with
[2] A map f: A → X of topological spaces is a (Hurewicz) cofibration if it has the homotopy extension property for maps to any space.
A cofibration f is always injective, in fact a homeomorphism to its image.
The inclusion of a closed subspace A in a space X is a cofibration if and only if A is a neighborhood deformation retract of X, meaning that there is a continuous map
[4] For example, the inclusion of a subcomplex in a CW complex is a cofibration.
(See Brouwer fixed-point theorem § A proof using homology or cohomology.)
such as normal spaces have been considered in this definition, but the class
of metrizable spaces has been found to give the most satisfactory theory.
For that reason, the notations AR and ANR by themselves are used in this article to mean
[6] A metrizable space is an AR if and only if it is contractible and an ANR.
[7] By Dugundji, every locally convex metrizable topological vector space
is an AR; more generally, every nonempty convex subset of such a vector space
[8] For example, any normed vector space (complete or not) is an AR.
ANRs form a remarkable class of "well-behaved" topological spaces.