locally EuclideanLocallyEuclidean2004-03-12 11:14:432007-03-19 15:10:18Definitionlocally Euclidean spacechart% this is the default PlanetMath preamble. as your knowledge
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\newcommand*{\abs}[1]{| #1 |}A locally Euclidean space $X$ is a topological space that locally
``looks'' like $\R^n$.
This makes it possible to talk about
coordinate axes around $X$. It also gives some topological structure
to the space: for example, since $\R^n$ is locally compact, so is $X$.
However, the restriction does not induce any geometry onto $X$.
\PMlinkescapeword{constant}
{\bf Definition}
Suppose $X$ is a topological space. Then $X$ is
called \emph{locally Euclidean} if for each $x\in X$ there is a neighbourhood
$U\subseteq X$, a $V\subseteq \sR^n$, and
a homeomorphism $\phi: U\to V$. Then the triple $(U,\phi, n)$
is called a \emph{chart} for $X$.
Here, $\sR$ is the set of real numbers, and for $n=0$ we define
$\sR^0$ as set with a single point equipped with the discrete topology.
\subsubsection*{Local dimension}
Suppose $X$ is a locally Euclidean space with $x\in X$. Further,
suppose $(U,\phi, n)$ is a chart of $X$ such that $x\in U$.
Then we define the \emph{local \PMlinkescapetext{dimension}} of $X$ at $x$ is $n$.
This is well defined, that is, the local dimension does not
depend on the chosen chart. If
$(U',\phi', n')$ is another chart with $x\in U'$, then
$\psi\circ \phi^{-1}: \phi(U\cap U') \to \psi(U\cap U')$
is a homeomorphism between $\phi(U\cap U')\subseteq \sR^n$
and $\psi(U\cap U')\subseteq \sR^{n'}$. By Brouwer's theorem
for the invariance of dimension (which is nontrivial),
it follows that $n=n'$.
If the local dimension is constant, say $n$, we say that the dimension
of $X$ is $n$, and write $\dim X = n$.
\subsubsection*{Examples}
\begin{itemize}
\item Any set with the discrete topology, is a locally
Euclidean of dimension $0.$
\item Any open subset of $\sR^n$ is locally Euclidean.
\item Any manifold is locally Euclidean. For example,
using a stereographic projection, one can show that the sphere $S^n$
is locally Euclidean.
\item The long line is locally Euclidean of dimension one. Note that the long line is not Hausforff. \cite{conlon}.
\end{itemize}
\subsubsection*{Notes}
The concept locally Euclidean has a different meaning in the
setting of Riemannian manifolds.
\begin{thebibliography}{9}
\bibitem {conlon} L. Conlon, \emph{Differentiable Manifolds: A first course},
Birkh\"auser, 1993.
\end{thebibliography}