<abstract xmlns="http://www.w3.org/1999/xhtml"><p> In this article we focus on a special case of the Brouwer invariance of domain theorem. Let us A, B be a subsets of ε<sup>n</sup>, 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.</p></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.

Brouwer Invariance of Domain Theorem

Institute of Informatics University of Białystok Sosnowa 64, 15-887 Białystok Poland

Formalized Mathematics

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. (CC BY-SA 3.0)

{"article-title":"Brouwer Invariance of Domain Theorem"}

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...

Brouwer Invariance of Domain Theorem

Published Online: Mar 30, 2014

Page range: 21 - 28

DOI: https://doi.org/10.2478/forma-2014-0003

Keywords
continuous transformations, topological dimension

© by Karol Pąk

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. (CC BY-SA 3.0)

Brouwer Invariance of Domain Theorem

Published Online: Mar 30, 2014

Page range: 21 - 28

DOI: https://doi.org/10.2478/forma-2014-0003

Keywordscontinuous transformations, topological dimension

© by Karol Pąk

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. (CC BY-SA 3.0)

Keywords
continuous transformations, topological dimension