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Concept
Invariance of domain
Summary
Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space .
It states:
If is an open subset of and is an injective continuous map, then is open in and is a homeomorphism between and .
The theorem and its proof are due to L. E. J. Brouwer, published in 1912.
The proof uses tools of algebraic topology, notably the Brouwer fixed point theorem.
The conclusion of the theorem can equivalently be formulated as: " is an open map".
Normally, to check that is a homeomorphism, one would have to verify that both and its inverse function are continuous;
the theorem says that if the domain is an subset of and the image is also in then continuity of is automatic.
Furthermore, the theorem says that if two subsets and of are homeomorphic, and is open, then must be open as well.
(Note that is open as a subset of and not just in the subspace topology.
Openness of 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 of are contained in Euclidean space .
Consider for instance the map defined by
This map is injective and continuous, the domain is an open subset of , but the image is not open in
A more extreme example is the map defined by because here 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.
Define as the shift
Then is injective and continuous, the domain is open in , but the image is not.
An important consequence of the domain invariance theorem is that cannot be homeomorphic to if
Indeed, no non-empty open subset of can be homeomorphic to any open subset of in this case.
The domain invariance theorem may be generalized to manifolds: if and are topological n-manifolds without boundary and is a continuous map which is locally one-to-one (meaning that every point in has a neighborhood such that restricted to this neighborhood is injective), then is an open map (meaning that is open in whenever is an open subset of ) and a local homeomorphism.
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In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.
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