Invariance of domain

Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space

It states: The theorem and its proof are due to L. E. J. Brouwer, published in 1912.

[1] The proof uses tools of algebraic topology, notably the Brouwer fixed point theorem.

The conclusion of the theorem can equivalently be formulated as: "

are continuous; the theorem says that if the domain is an open subset of

and not just in the subspace topology.

in the subspace topology is automatic.)

Both of these statements are not at all obvious and are not generally true if one leaves Euclidean space.

It is of crucial importance that both domain and image of

are contained in Euclidean space of the same dimension.

This map is injective and continuous, the domain is an open subset of

is injective and continuous but does not even yield a homeomorphism onto its image.

The theorem is also not generally true in infinitely many dimensions.

Consider for instance the Banach Lp space

of all bounded real sequences.

is injective and continuous, the domain is open in

, there exists no continuous injective map

for a nonempty open set

To see this, suppose there exists such a map

would give a continuous injection from

to itself, but with an image with empty interior in

This would contradict invariance of domain.

, no nonempty open subset of

can be homeomorphic to an open subset of

The domain invariance theorem may be generalized to manifolds: if

are topological n-manifolds without boundary and

is a continuous map which is locally one-to-one (meaning that every point in

restricted to this neighborhood is injective), then

is an open map (meaning that

There are also generalizations to certain types of continuous maps from a Banach space to itself.