Invariance Of Domain

(Theory)

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What is 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. The proof uses tools of algebraic topology, notably the Brouwer fixed point theorem.

Technology Types

algebraic topologyhomeomorphismmathematical theorempropositionstatementtheoremtheorems in topologytheory

Synonyms

Brouwer's theorem on domain invarianceDomain invariance theorem

Translations

Invarianza del dominio (es)Théorème de l'invariance du domaine (fr)Twierdzenie Brouwera o zachowaniu otwartości (pl)Теорема об инвариантности области (ru)

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