@MISC{BROUWER3.MIZ,
  AUTHOR = {P\k{a}k, Karol},
  TITLE = {Brouwer Invariance of Domain Theorem},
  NOTE = {The paper has been financed by the
  resources of the Polish National Science Centre granted by decision
  no DEC-2012/07/N/ST6/02147.},
  SECTION1 = {Preliminaries},
  SECTION2 = {A Distribution of Sphere},
  SECTION3 = {A Characterization of Open Sets in Euclidean Space in Terms of Continuous Transformations},
  SECTION4 = {Brouwer Invariance of Domain Theorem -- Special Case},
  SECTION5 = {Topological Invariance of Dimension -- An Introduction to Manifolds},
  DAY = {11},
  MONTH = {February},
  YEAR = {2014},
  ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok},
  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 ${\cal E}^n$, and $f:A \rightarrow B$ 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 a basic properties of the $n$-dimensional manifolds, and also
  to prove a 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 \cite{DUDA:BM61};
             \cite{Engelking:1989},
             \cite{Engelking:1978} can also serve as reference books.}}
