T1 spaceT1Space2002-02-08 18:14:412005-05-18 03:13:21Definition%\usepackage{graphicx}
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\newcommand{\mathbb}[1]{\mathbbmss{#1}}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 $y\notin U$.
A space being $T_1$ is equivalent to the following statements:
\begin{itemize}
\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 Distinct points are separated.
\end{itemize}