Invariance Of Dimension
Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space \R^n. It states: :If U is an open subset of \R^n and f : U \rarr \R^n is an injective continuous map, then V := f(U) is open in \R^n and f is a homeomorphism between U and V. 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. 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^ are continuous; the theorem says that if the domain is an subset of \R^n and the image is also in \R^n, then continuity of f^ 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. (Note that V is open as a subset of \R^n, and not just in the subspace topology. Openness ...
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Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Twist (mathematics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity (mathematics), continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopy, homotopies. A property that is invariant under such deformations is a topological property. Basic exampl ...
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