Invariance of domain and retraction

Invariacne of domain

Invariance of domain states:

Theorem (Invariance of domain) If U is an open subset of \mathbf{R}^n and f:U\to\mathbf{R}^n is an injective continuous map, then V=f(U) is open and f is a homeomorphism between U and V.

The conclusion of the theorem can equivalently be formulated as: “f is an open map”. It is of crucial importance that both domain and range of f are contained in the Euclidean space of the same dimension. The theorem is also not generally true in infinite dimension.

retraction is a continuous mapping from the entire space into a subspace which preserves the position of all points in that space. A deformation retraction is a map which captures the idea of continuously shrinking a space into a subspace.

Retract

Definition Let X be a topological space and A be a subspace of X. Then a continuous map

\displaystyle r:X\to A

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

\displaystyle \iota:A\hookrightarrow X

the inclusion, a retraction is a continuous map r such that

\displaystyle r\circ\iota=\mathrm{id}_A.

A subspace A is called a retract of X if such a retraction exists.

Theorem Let X be a Hausdorff space and A be a retract of X. Then A is closed.

Proof Let x\notin A and a=r(x)\in A. Since X is Hausdorff, x and a have disjoint neighborhood U and V, respectively. Then r^{-1}(V\cap A)\cap U is a neighborhood of x disjoint from A. Hence, A is closed. \Box

A space X is known as an absolute retract if for every normal space Y contains X4 as a closed subspace, latex X$ is a retract of Y.

Deformation retract and strong deformation retract

Definition A continuous map

\displaystyle F:X\times [0,1]\to X

is a deformation retraction of a space X onto a subspace A if, for every x in X and a in A,

\displaystyle F(x,0)=x,\ F(x,1)\in A,\ \text{and}\ F(a,1)=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 A. A deformation retraction is a special case of homotop equivalence.

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