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.
@article{bwmeta1.element.doi-10_2478_forma-2014-0003,
author = {Karol P\k ak},
title = {Brouwer Invariance of Domain Theorem},
journal = {Formalized Mathematics},
volume = {22},
year = {2014},
pages = {21-28},
zbl = {1298.54004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_forma-2014-0003}
}
Karol Pąk. Brouwer Invariance of Domain Theorem. Formalized Mathematics, Tome 22 (2014) pp. 21-28. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_forma-2014-0003/
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