54-00-NiemytzkiPlane.tex

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\pmtitle{Niemytzki plane}
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\pmtype{Example}
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\pmsynonym{Niemytzki space}{NiemytzkiPlane}

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\begin{document}
Let $\Gamma$ be the Euclidean half plane $\Gamma =\{(x,y)\mid y\ge0\}\sse\R^2$,
with the usual subspace topology. We enrich the topology on $\Gamma$ by throwing
in open sets of the form $\{(x,0)\}\cup B_r(x,r)$, that is an open ball of radius
$r$ around $(x,r)$ together with its point tangent to $\R\times\{0\}$
(Fig.~\ref{nbhd}).
\begin{figure}[h]
\begin{center}
  \includegraphics{nbhd}
  \caption{A new open set in $\Gamma$. It consists of an open
    disc and the point tangent to $y=0$.\label{nbhd}}
\end{center}
\end{figure}
The space $\Gamma$ endowed with the enriched topology is called the
\emph{Niemytzki plane}.

Some miscellaneous properties of the Niemytzki plane are
\begin{itemize}
\item the subspace $\R\times\{0\}$ of $\Gamma$ is discrete, hence
  the only convergent sequences in this subspace are constant ones;
\item it is Hausdorff;
\item it is completely regular;
\item it is not normal.
\end{itemize}
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\end{document}