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.
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.