Let X be a topological space and A a subspace of X. Then a continuous map
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 space retracts to a point in the obvious way (the constant map yields a retraction).
A space X is known as an absolute retract (or AR) if for every normal space Y that embeds X as a closed subset, X is a retract of Y.
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.
A continuous map
is a deformation retraction if, for every x in X and a in A,
In other words, a deformation retraction is a homotopy between a retract and the identity map on X. The subspace A is called a deformation retract of X. A deformation retract 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], d is called a strong deformation retraction. In other words, a strong deformation retraction leaves points in A fixed throughout the homotopy. (Some authors take this as the definition of deformation retraction.)
A pair of spaces in U is an NDR-pair if there exists a map such that and a homotopy such that for all , for all , and for all . The pair is said to be a representation of as an NDR-pair.
Any topological space which deformation retracts to a point is contractible and vice versa. However, there exist contractible spaces which do not strongly deformation retract to a point.