出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2015/07/31 09:28:50」(JST)
In topology, a branch of mathematics, a retraction[1] is a continuous mapping from the entire space into a subspace which preserves the position of all points in that subspace. A deformation retraction is a map which captures the idea of continuously shrinking a space into a subspace.
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, r(a) = a 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 closed.
If is a retraction, then the composition 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 space X is known as an absolute retract if for every normal space Y that contains X as a closed subspace, X is a retract of Y. The unit cube In as well as the Hilbert cube Iω are absolute retracts.
If there exists an open set U such that
and A is a retract of U, then A is called a neighborhood retract of X.
A space X is an absolute neighborhood retract (or ANR) if for every normal space Y that embeds X as a closed subset, X is a neighborhood retract of Y. The n-sphere Sn is an absolute neighborhood retract.
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 homotopy equivalence.
A retract need not be a deformation retract. For instance, having a single point as a deformation retract would imply a space is path connected (in fact, it would imply contractibility of the space).
Note: An equivalent definition of deformation retraction is the following. A continuous map r: X → A is a deformation retraction if it is a retraction and its composition with the inclusion is homotopic to the identity map on X. In this formulation, a deformation retraction carries with it a homotopy between the identity map on X and itself.
If, in the definition of a deformation retraction, we add the requirement that
for all t in [0, 1] and a in A, then F is called a strong deformation retraction. In other words, a strong deformation retraction leaves points in A fixed throughout the homotopy. (Some authors, such as Allen Hatcher, take this as the definition of deformation retraction.)
As an example, the n-sphere Sn is a strong deformation retract of Rn+1\{0}; as strong deformation retraction one can choose the map
A closed subspace A is a neighborhood deformation retract of X if there exists a continuous map (where ) such that and a homotopy such that for all , for all , and for all .[2]
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