Retract

Retract of topological space $X$ ${\ displaystyle X}$ $X$ - subspace $A$ ${\ displaystyle A}$ $A$ this space for which there is a retraction $X$ ${\ displaystyle X}$ $X$ on $A$ ${\ displaystyle A}$ $A$ ; i.e. continuous mapping $f:X\to A$ ${\ displaystyle f: X \ to A}$ ${\ displaystyle f: X \ to A}$ identical to $A$ ${\ displaystyle A}$ $A$ (i.e. such that $f(x)=x$ ${\ displaystyle f (x) = x}$ $f (x) = x$ for all $x\in A$ ${\ displaystyle x \ in A}$ $x \ in A$ ).

The retract of a topological space inherits many important properties of the space itself, at the same time it can be arranged much simpler than itself, more observable, more convenient for a specific study.

Content

Examples

A one-point set is a retract of a segment, a straight line, a plane, etc.
Every non-empty closed set of a Cantor perfect set is its retract.
$n$ ${\ displaystyle n}$ -dimensional sphere is not a retract $(n+1)$ ${\ displaystyle (n + 1)}$ -dimensional ball of Euclidean space, since the ball has zero homology groups , and the sphere is a non-zero group $H_{n}$ ${\ displaystyle H_ {n}}$ . This contradicts the existence of a retract, since retraction induces an epimorphism of homology groups.

Related definitions

Subspace $A$ ${\ displaystyle A}$ spaces $X$ ${\ displaystyle X}$ called a neighborhood retract if in $X$ ${\ displaystyle X}$ there is an open subspace containing $A$ ${\ displaystyle A}$ whose retract is $A$ ${\ displaystyle A}$ .
Metrizable space $X$ ${\ displaystyle X}$ is called an absolute retract ( absolute neighborhood retract ) if it is a retract (respectively, a neighborhood retract) of every metrizable space containing $X$ ${\ displaystyle X}$ as a closed subspace.
If retraction of space $X$ ${\ displaystyle X}$ on its subspace $A$ ${\ displaystyle A}$ homotopic to the identity mapping of space $X$ ${\ displaystyle X}$ on myself then $A$ ${\ displaystyle A}$ called deformation retract of space $X$ ${\ displaystyle X}$ .
Linear operator $P$ ${\ displaystyle P}$ in topological vector space $E$ ${\ displaystyle E}$ , which is a retraction, is called a continuous projector . Vector subspace $F$ ${\ displaystyle F}$ topological vector space $E$ ${\ displaystyle E}$ called complemented if there is a continuous projector $P\colon E\to F$ ${\ displaystyle P \ colon E \ to F}$ .

Properties

Subspace $A$ ${\ displaystyle A}$ spaces $X$ ${\ displaystyle X}$ is its retract if and only if every continuous mapping of space $A$ ${\ displaystyle A}$ in arbitrary topological space $Y$ ${\ displaystyle Y}$ can continue to display the entire space continuously $X$ ${\ displaystyle X}$ at $Y$ ${\ displaystyle Y}$ .
If space $X$ ${\ displaystyle X}$ - Hausdorff , then every retract of space $X$ ${\ displaystyle X}$ closed in $X$ ${\ displaystyle X}$ .
Any property that remains in the transition to a continuous image, as well as any property inherited by closed subspaces, is stable with respect to the transition to a retract. In particular, during the transition to the retract remain
- compactness
- connectivity
- linear connectivity
- separability ,
- the upper limit on the dimension ,
- paracompactness
- normality
- local compactness
- local connectivity .
If space $X$ ${\ displaystyle X}$ has the property of a fixed point , i.e. for each continuous display $f:X\to X$ ${\ displaystyle f: X \ to X}$ there is a point $x\in X$ ${\ displaystyle x \ in X}$ such that $f(x)=x$ ${\ displaystyle f (x) = x}$ , then every retract of space $X$ ${\ displaystyle X}$ has the property of a fixed point.
An absolute neighborhood retract is a locally contractible space .
Retraction induces epimorphism of homology groups .

Literature

Borsuk K., Theory of Retracts, trans. from English, M., 1971.