completely HausdorffCompletelyHausdorff2004-03-17 13:14:312007-09-14 09:52:23Definition\usepackage{graphicx}
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\newtheorem{cor}{Corollary}\begin{defn} \cite{steen}
Let $(X,\tau)$ be a topological space.
Suppose that for any two different points $x,y\in X, x\neq y$,
we can find two disjoint neighborhoods
\[U_x,V_y\in \tau,\qquad x\in U_x, y\in Y_y\] such that their
closures are also disjoint:
\[\overline{U_x}\cap \overline{V_y}=\emptyset.\]
Then we say that $(X,\tau)$ is a
\emph{completely Hausdorff space} or a \emph{$T_{2\frac12}$ space}.
\end{defn}
\subsubsection*{Notes}
A synonym for functionally Hausdorff space is
\emph{Urysohn space} \cite{steen}.
Unfortunately, the definition of completely Hausdorff and $T_{2\frac12}$
are not as standard as one would like since. For example, the
term completely Hausdorff space is also used to mean
a functionally Hausdorff space (e.g. \cite{willard}).
Nevertheless, in the present convention, we have the implication:
\[
\mbox{functionally Hausdorff}
\Rightarrow
\mbox{completely Hausdorff}
\Rightarrow
T_2=\mbox{Hausdorff},
\]
which suggests why the $T_{2\frac12}$ name have been used to
denote both completely Hausdorff spaces and functionally Hausdorff spaces.
\begin{thebibliography}{9}
\bibitem{steen} L.A. Steen, J.A.Seebach, Jr.,
\emph{Counterexamples in topology},
Holt, Rinehart and Winston, Inc., 1970.
\bibitem{willard} S. Willard, \emph{General Topology},
Addison-Wesley Publishing Company, 1970.
\end{thebibliography}