Retract of a topological space

A subspace $ A $ of a topological space $ X $ for which there is a retraction of $ X $ onto $ A $. If $ X $ is a Hausdorff space, then every retract of $ X $ is closed in $ X $. Every non-empty closed subset of the Cantor perfect set is a retract of it. In the transition from a space to a retract of it many important properties are preserved. In particular, every property which is preserved under transition to a continuous image is, like any property of Hausdorff spaces inherited by closed subspaces, stable under passage to a retract. As a result, compactness, connectedness, path-connectedness, separability, an upper bound on the dimension, paracompactness, normality, local compactness, and local connectedness are preserved under passage to a retract. At the same time, a retract of a space may have a simpler structure than the space itself, and be more amenable and convenient for specific research. Thus, a one-point set is a retract of an interval, of a straight line, of a plane, etc. If the space $ X $ has the fixed point property, i.e. if for each continuous transformation $ f: X \rightarrow X $ there is a point $ x \in X $ such that $ f( x) = x $, then each retract of $ X $ possesses the fixed point property too. In particular, the $ n $- dimensional sphere is not a retract of the $ ( n+ 1) $- dimensional ball of a Euclidean space, where $ n= 0, 1 \dots $ since the closed ball has the fixed point property (Brouwer's fixed-point theorem), and the sphere does not. A subspace $ A $ of a space $ X $ is called a neighbourhood retract of this space if there is in $ X $ an open subspace which contains $ A $ and of which $ A $ is a retract. The concept of a retract is directly related to the problem of the extension of continuous mappings. Thus, a subspace $ A $ is a retract of $ X $ if and only if every continuous mapping of $ A $ into an arbitrary topological space $ Y $ can be extended to a continuous mapping of the entire space $ X $ into $ Y $.

A metric space $ X $ is called an absolute retract (absolute neighbourhood retract) if it is a retract (neighbourhood retract) of every metric space containing $ X $ as a closed subspace. For a metric space $ X $ to be an absolute retract it is necessary that it be a retract of some convex subspace of a normed linear space, and it is sufficient that $ X $ be a retract of a convex subspace of a locally convex linear space.

Thus, all convex subspaces of locally convex linear spaces are absolute retracts; such is the case, in particular, with a point, an interval, a ball, a straight line, etc. This characterization means that absolute retracts have the following properties. Every retract of an absolute retract is again an absolute retract. Each absolute retract is contractible in itself and is locally contractible. All homology, cohomology, homotopy, and cohomotopy groups of an absolute retract are trivial. A metric space $ Y $ is an absolute retract if and only if, given any metric space $ X $, a closed subspace $ A $ of $ X $ and a continuous mapping of $ A $ into $ Y $, the mapping can be extended to a continuous mapping of the entire space $ X $ into $ Y $. Absolute neighbourhood retracts are characterized as retracts of open subsets of convex subspaces of normed linear spaces. They include all compact polyhedra. An important property of them is their local contractibility.

If a retraction of a space $ X $ into a subspace $ A $ of $ X $ is homotopic to the identity mapping of $ X $ into itself, then $ A $ is called a deformation retract of $ X $. A deformation retract of a space $ X $ is homotopy equivalent to $ X $, i.e. they have the same homotopy type. Conversely, two homotopy-equivalent spaces can always be imbedded in a third space in such a way that they are both deformation retracts of this space.

References

[1]	K. Borsuk, "Theory of retracts" , PWN (1967)

Comments

"Absolute retract" and "absolute neighbourhood retract" are often abbreviated to AR and ANR.

Retracts and absolute retracts have been studied in other classes of spaces than the metrizable ones, most successfully in compact Hausdorff spaces and in $ T _ {0} $- spaces. A compact Hausdorff absolute retract is the same thing as a retract of a Tikhonov cube. If such a space is finite-dimensional (in the sense of covering dimension), it is metrizable, [a1]. The $ T _ {0} $- absolute retracts or injective $ T _ {0} $- spaces have a natural partial ordering which makes them continuous lattices (cf. Continuous lattice).

References

[a1]	E.S. Shchepin, "A finite-dimensional compact absolute neighborhood retract is metrizable" Soviet Math. Doklady , 18 (1977) pp. 402–406 Dokl. Akad. Nauk SSSR , 233 : 3 (1977) pp. 304–307
[a2]	J. van Mill, "Infinite-dimensional topology, prerequisites and introduction" , North-Holland (1988)