### Invariacne of domain

*Invariance of domain *states:

**Theorem (Invariance of domain) **If is an open subset of and is an injective continuous map, then is open and is a homeomorphism between and .

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

A *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 be a topological space and be a subspace of . Then a continuous map

is a *retraction* if the restriction of to is the identity map on ; that is, for all in . Equivalently, denoting by

the inclusion, a retraction is a continuous map such that

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

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

*Proof *Let and . Since is Hausdorff, and have disjoint neighborhood and , respectively. Then is a neighborhood of disjoint from . Hence, is closed.

A space is known as an *absolute retract* if for every normal space contains latex X$ is a retract of .

### Deformation retract and strong deformation retract

**Definition **A continuous map

is a *deformation retraction *of a space onto a subspace if, for every in and in ,

In other words, a deformation retraction is a homotopy between a retraction and the identity map on . The subspace is called a *deformation retract *of . A deformation retraction is a special case of homotop equivalence.