54-00-LocallyHomeomorphic.tex

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\pmtitle{locally homeomorphic}
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\pmtype{Definition}
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\pmsynonym{local homeomorphy}{LocallyHomeomorphic}
\pmrelated{LocallyEuclidean}

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\begin{document}
Let $X$ and $Y$ be topological spaces. Then $X$ is \emph{locally
  homeomorphic} to $Y$, if for every $x\in X$ there is a neighbourhood
  $U\subseteq X$ of $x$ and an \PMlinkid{open set}{380} $V\subseteq Y$, such that $U$
  and $V$ with their respective subspace topology are homeomorphic.

\subsubsection*{Examples}
\begin{itemize}
\item Let $X=\{1\}$ and $Y=\{2,3\}$ be discrete spaces with one resp.\
  two elements. Since $X$ and $Y$ have different cardinalities,
  they cannot be homeomorphic. They are, however, locally homeomorphic
  to each other.
\item Again, let $X=\{1\}$ be a discrete space with one element, but
  now let $Y=\{2,3\}$ the space with topology
  $\{\emptyset,\{2\},Y\}$. Then $X$ is still locally homeomorphic to
  $Y$, but $Y$ is not locally homeomorphic to $X$, since the smallest
  neighbourhood of $3$ already has more elements than $X$.
\item Now, let $X$ be as in the previous examples, and $Y=\{2,3\}$
  be \PMlinkid{indiscrete}{3120}. Then neither $X$ is locally homeomorphic to $Y$ nor
  the other way round.
\item Non-trivial examples arise with locally Euclidean spaces,
  especially manifolds.
\end{itemize}
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