Brouwer Invariance of Domain Theorem

Karol Pąk

Formalized Mathematics (2014)

  • Volume: 22, Issue: 1, page 21-28
  • ISSN: 1426-2630

Abstract

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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.

How to cite

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Karol Pąk. "Brouwer Invariance of Domain Theorem." Formalized Mathematics 22.1 (2014): 21-28. <http://eudml.org/doc/266703>.

@article{KarolPąk2014,
abstract = {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.},
author = {Karol Pąk},
journal = {Formalized Mathematics},
keywords = {continuous transformations; topological dimension},
language = {eng},
number = {1},
pages = {21-28},
title = {Brouwer Invariance of Domain Theorem},
url = {http://eudml.org/doc/266703},
volume = {22},
year = {2014},
}

TY - JOUR
AU - Karol Pąk
TI - Brouwer Invariance of Domain Theorem
JO - Formalized Mathematics
PY - 2014
VL - 22
IS - 1
SP - 21
EP - 28
AB - 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.
LA - eng
KW - continuous transformations; topological dimension
UR - http://eudml.org/doc/266703
ER -

References

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