| Version current |
Version 19 |
| A topological space $(X,\tau)$ is said to be $T_2$ |
A topological space $(X,\tau)$ is said to be $T_2$ |
| (or said to satisfy the $T_2$ axiom) if given |
(or said to satisfy the $T_2$ axiom) if given |
| distinct $x,y\in X$, there exist disjoint |
distinct $x,y\in X$, there exist disjoint |
| open sets $U,V\in\tau$ (that is, $U\cap V=\emptyset$) |
open sets $U,V\in\tau$ (that is, $U\cap V=\emptyset$) |
| such that $x\in U$ and $y\in V$. |
such that $x\in U$ and $y\in V$. |
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| A $T_2$ space is also known as a \emph{Hausdorff space}. |
A $T_2$ space is also known as a \emph{Hausdorff space}. |
| A \emph{Hausdorff topology} for a set $X$ is a topology |
A \emph{Hausdorff topology} for a set $X$ is a topology |
| $\tau$ such that $(X,\tau)$ is a Hausdorff space. |
$\tau$ such that $(X,\tau)$ is a Hausdorff space. |
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| \subsubsection*{Properties} |
\subsubsection*{Properties} |
| The following properties are equivalent: |
The following properties are equivalent: |
| \begin{enumerate} |
\begin{enumerate} |
| \item $X$ is a Hausdorff space. |
\item $X$ is a Hausdorff space. |
| \item The set |
\item The set |
| $$ |
$$ |
| \Delta=\{(x,y)\in X\times X:x=y\} |
\Delta=\{(x,y)\in X\times X:x=y\} |
| $$ |
$$ |
| is closed in the product topology of $X\times X$. |
is closed in the product topology of $X\times X$. |
| \item For all $x\in X$, we have |
\item For all $x\in X$, we have |
| $$ |
$$ |
| \{x\} = \bigcap \{A : A\subseteq X\ \mbox{closed}, \mbox{$\exists$ open set}\ U\ \mbox{such that}\ x\in U\subseteq A\}. |
\{x\} = \bigcap \{A : A\subseteq X\ \mbox{closed}, \mbox{$\exists$ open set}\ U\ \mbox{such that}\ x\in U\subseteq A\}. |
| $$ |
$$ |
| \end{enumerate} |
\end{enumerate} |
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| Important examples of Hausdorff spaces are metric spaces, manifolds, |
Important examples of Hausdorff spaces are metric spaces, manifolds, |
| and topological vector spaces. |
and topological vector spaces. |
