# Brouwer Invariance of Domain Theorem

Formalized Mathematics (2014)

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

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topKarol 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 -

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