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A topological space $(X,\tau)$ is said to be $T_1$ (or said to hold the $T_1$ axiom) if for all distinct points $x,y\in X$ ($x\neq y$), there exists an open set $U\in\tau$ such that $x\in U$ and $y\notin V$.
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A topological space $(X,\tau)$ is said to be $T_1$ (or said to hold the $T_1$ axiom) if given $x,y\in X$ ($x\neq y$), there exist open sets $U,V\in\tau$ such that $x\in U$ and $y\in V$.
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| A space being $T_1$ is equivalent to the following statements: |
A space being $T_1$ is equivalent to the following statements: |
| \begin{itemize} |
\begin{itemize} |
| \item For every $x\in X$, the set $\{x\}$ is closed. |
\item For every $x\in X$, the set $\{x\}$ is closed. |
| \item Every subset of $X$ is equal to the intersection of all the open sets that contain it. |
\item Every subset of $X$ is equal to the intersection of all the open sets that contain it. |
| \end{itemize} |
\end{itemize} |
