- Invariance of domain
**Invariance of domain**is a theorem intopology abouthomeomorphic subset s ofEuclidean space **R**^{"n"}. It states: :If "U" is an open subset of**R**^{"n"}and "f" : "U" →**R**^{"n"}is aninjective continuous map , then "V" = "f"("U") is open and "f" is ahomeomorphism between "U" and "V".The theorem and its proof are due to

L.E.J. Brouwer , published in 1912. [*Brouwer L. Zur Invarianz des "n"-dimensionalen Gebiets, "*] The proof uses tools ofMathematische Annalen " 72 (1912), pages 55 - 56algebraic topology , notably theBrouwer 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**R**^{"n"}and the image is also in**R**^{"n"}, then continuity of "f"^{ -1}is automatic. Furthermore, the theorem says that if two subsets "U" and "V" of**R**^{"n"}are homeomorphic, and "U" is open, then "V" must be open as well. 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 range of "f" are contained in Euclidean space "of the same dimension". Consider for instance the map "f" : (0,1) →

**R**^{2}with "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**R**^{2}. A more extreme example is "g" : (-1.1,1) →**R**^{2}with "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 realsequence s. Define "f" : "l"^{∞}→ "l"^{∞}as the shift "f"("x"_{1},"x"_{2},...) = (0, "x"_{1}, "x"_{2},...). 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

**R**^{"n"}cannot be homeomorphic to**R**^{"m"}if "m" ≠ "n". Indeed, no non-empty open subset of**R**^{"n"}can be homeomorphic to any open subset of**R**^{"m"}in this case. (Proof: If "m" < "n", then we can view**R**^{"m"}as a subspace of**R**^{"n"}, and the non-empty open subsets of**R**^{"m"}are not open when considered as subsets of**R**^{"n"}. We apply the theorem in the space**R**^{"n"}.)...**Generalizations**The domain invariance theorem may be generalized to

manifold s: 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 anopen map (meaning that "f"("U") is open in "N" whenever "U" is an open subset of "M").There are also generalizations to certain types of continuous maps from a

Banach space to itself. [*Leray J. Topologie des espaces abstraits de M. Banach. "C.R. Acad. Sci. Paris", 200 (1935) pages 1083–1093*]**ee also***

Open mapping theorem for other conditions that ensure that a given continuous map is open.**References****ources*** [

*http://eom.springer.de/D/d120250.htm Domain Invariance*] , from theEncyclopaedia of Mathematics

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