Brouwer Invariance of Domain Theorem
| Autor: | Karol Pąk |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2014 |
| Předmět: |
continuous transformations
Applied Mathematics Dimension (graph theory) Boundary (topology) Manifold Combinatorics Computational Mathematics symbols.namesake Invariance of domain symbols QA1-939 topological dimension Special case Focus (optics) Lebesgue covering dimension Degree of a continuous mapping Mathematics |
| Zdroj: | Formalized Mathematics, Vol 22, Iss 1, Pp 21-28 (2014) |
| ISSN: | 1898-9934 |
| Popis: | Summary In this article we focus on a special case of the Brouwer invariance of domain theorem. Let us A, B be a subsets of εn, and f : A → B be a homeomorphic. We prove that, if A is closed then f transform the boundary of A to the boundary of B; and if B is closed then f transform the interior of A to the interior of B. These two cases are sufficient to prove the topological invariance of dimension, which is used to prove basic properties of the n-dimensional manifolds, and also to prove basic properties of the boundary and the interior of manifolds, e.g. the boundary of an n-dimension manifold with boundary is an (n − 1)-dimension manifold. This article is based on [18]; [21] and [20] can also serve as reference books. |
| Databáze: | OpenAIRE |
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