# 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.