Invariance of domain
Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space ℝn. It states:
- If U is an open subset of ℝn and f : U → ℝn is an injective continuous map, then V := f(U) is open in ℝn and f is a homeomorphism between U and V.
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.
Notes
The conclusion of the theorem can equivalently be formulated as: "f is an open map".
Normally, to check that f is a homeomorphism, one would have to verify that both f and its inverse function f −1 are continuous; the theorem says that if the domain is an open subset of ℝn and the image is also in ℝn, then continuity of f −1 is automatic. Furthermore, the theorem says that if two subsets U and V of ℝn are homeomorphic, and U is open, then V must be open as well. (Note that V is open as a subset of ℝn, and not just in the subspace topology. Openness of V 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.
![](../I/m/A_map_which_is_not_a_homeomorphism_onto_its_image.png)
It is of crucial importance that both domain and range of f are contained in Euclidean space of the same dimension. Consider for instance the map f : (0,1) → ℝ2 defined by f(t) = (t, 0). This map is injective and continuous, the domain is an open subset of ℝ, but the image is not open in ℝ2. A more extreme example is the map g : (−1.1, 1) → ℝ2 defined by g(t) = (t 2 − 1, t 3 − t) because here g is injective and continuous but does not even yield a homeomorphism onto its image.
The theorem is also not generally true in infinite dimensions. Consider for instance the Banach space l∞ of all bounded real sequences. Define f : l∞ → l∞ as the shift f(x1, x2, ...) = (0, x1, x2, ...). Then f is injective and continuous, the domain is open in l∞, but the image is not.
Consequences
An important consequence of the domain invariance theorem is that ℝn cannot be homeomorphic to ℝm if m ≠ n. Indeed, no non-empty open subset of ℝn can be homeomorphic to any open subset of ℝm in this case.
Generalizations
The domain invariance theorem may be generalized to manifolds: if M and N are topological n-manifolds without boundary and f : M → N is a continuous map which is locally one-to-one (meaning that every point in M has a neighborhood such that f restricted to this neighborhood is injective), then f is an open map (meaning that f(U) is open in N whenever U is an open subset of M) and a local homeomorphism.
There are also generalizations to certain types of continuous maps from a Banach space to itself.[2]
See also
- Open mapping theorem for other conditions that ensure that a given continuous map is open.
References
- Brouwer L.E.J. Beweis der Invarianz des n-dimensionalen Gebiets, Mathematische Annalen 71 (1912), pages 305–315; see also 72 (1912), pages 55–56
- Leray J. Topologie des espaces abstraits de M. Banach. C. R. Acad. Sci. Paris, 200 (1935) pages 1083–1093
External link s
- Mill, J. van (2001) [1994], "Domain invariance", Encyclopedia of Mathematics, EMS Press