Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space Rn. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Topological equivalence redirects here see also Topological equivalence (dynamical systems. 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 and f is a homeomorphism between U and V. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output Topological equivalence redirects here see also Topological equivalence (dynamical systems.
The theorem and its proof are due to L.E.J. Brouwer, published in 1912. Luitzen Egbertus Jan Brouwer ɛxˈbɛʁtəs jɑn ˈbʁʌuəʁ ( February 27 1881, Overschie – December 2 1966, Blaricum [1] The proof uses tools of algebraic topology, notably the Brouwer fixed point theorem. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic In Mathematics, the Brouwer fixed point theorem is an important Fixed point theorem that applies to finite-dimensional spaces and which forms the basis for several
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Notes
The conclusion of the theorem can equivalently be formulated as: "f is an open map". In Topology, an open map is a function between two Topological spaces which maps Open sets to open sets
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. In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B 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. 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. In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Mathematics, the range of a function is the set of all "output" values produced by that function Consider for instance the map f : (0,1) → R2 with f(t) = (t,0). In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set 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 g : (-1. 1,1) → R2 with g(t) = (t2-1, t3-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. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding In Mathematics, a sequence is an ordered list of objects (or events 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. (Proof: If m < n, then we can view Rm as a subspace of Rn, and the non-empty open subsets of Rm are not open when considered as subsets of Rn. We apply the theorem in the space Rn. ). . .
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). A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. In Topology, an open map is a function between two Topological spaces which maps Open sets to open sets
There are also generalizations to certain types of continuous maps from a Banach space to itself. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis [2]
See also
- Open mapping theorem for other conditions that ensure that a given continuous map is open.
References
- ^ Brouwer L. Zur Invarianz des n-dimensionalen Gebiets, Mathematische Annalen 72 (1912), pages 55 - 56
- ^ Leray J. The Mathematische Annalen (abbreviated as Math Ann or Math Annal Topologie des espaces abstraits de M. Banach. C. R. Acad. Sci. Paris, 200 (1935) pages 1083–1093
Sources
The Encyclopaedia of Mathematics is a large reference work in Mathematics.Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic