Retraction (topology)
Let "X" be a topological space and "A" a subspace of "X". Then a continuous map
formula_1
is a retraction if the restriction of "r" to "A" is the identity map on "A"; that is, formula_2 for all "a" in "A". Equivalently, denoting by
formula_3
the inclusion, a retraction is a continuous map "r" such that
formula_4
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 formula_5 is a retraction, then the composition ι∘"r" is an idempotent continuous map from "X" to "X". Conversely, given any idempotent continuous map formula_6 we obtain a retraction onto the image of "s" by restricting the codomain. A continuous map
formula_7
is a "deformation retraction" of a space "X" onto a subspace "A" if, for every "x" in "X" and "a" in "A",
formula_8
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. A continuous map formula_5 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
formula_10
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 Hatcher, take this as the definition of deformation retraction.) As an example, the "n"-sphere "formula_11" is a strong deformation retract of formula_12 as strong deformation retraction one can choose the map 
formula_13 A map "f": "A" → "X" of topological spaces is a (Hurewicz) cofibration if it has the homotopy extension property for maps to any space. This is one of the central concepts of homotopy theory. A cofibration "f" is always injective, in fact a homeomorphism to its image. If "X" is Hausdorff (or a compactly generated weak Hausdorff space), then the image of a cofibration "f" is closed in "X". Among all closed inclusions, cofibrations can be characterized as follows. 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 formula_14 with formula_15 and a homotopy formula_16 such that formula_17 for all formula_18 formula_19 for all formula_20 and formula_21 and formula_22 if formula_23. For example, the inclusion of a subcomplex in a CW complex is a cofibration. The boundary of the "n"-dimensional ball, that is, the ("n"−1)-sphere, is not a retract of the ball. (See .) A closed subset formula_28 of a topological space formula_29 is called a neighborhood retract of formula_29 if formula_28 is a retract of some open subset of formula_29 that contains formula_28.
Let formula_34 be a class of topological spaces, closed under homeomorphisms and passage to closed subsets. Following Borsuk (starting in 1931), a space "formula_28" is called an absolute retract for the class formula_34, written formula_37 if "formula_28" is in formula_34 and whenever "formula_28" is a closed subset of a space formula_29in formula_34, formula_28 is a retract of formula_29. A space formula_28 is an absolute neighborhood retract for the class formula_34, written formula_47 if formula_28 is in formula_34 and whenever formula_28 is a closed subset of a space formula_29 in formula_34, formula_28 is a neighborhood retract of formula_29. Various classes formula_34 such as normal spaces have been considered in this definition, but the class formula_56 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 formula_57 and formula_58. A metrizable space is an AR if and only if it is contractible and an ANR. By Dugundji, every locally convex metrizable topological vector space formula_59 is an AR; more generally, every nonempty convex subset of such a vector space formula_59 is an AR. For example, any normed vector space (complete or not) is an AR. More concretely, Euclidean space formula_61 the unit cube formula_62and the Hilbert cube formula_63 are ARs. ANRs form a remarkable class of "well-behaved" topological spaces. Among their properties are:
Definitions ; Deformation retract and strong deformation retract
A deformation retraction is a mapping that captures the idea of "continuously shrinking" a space into a subspace.