Definition 111.6.6. A topological space $X$ is said to verify the separation axiom $T_0$ if for any pair of points $x, y\in X$, $x\not= y$ there is an open subset of $X$ containing one but not the other. We say that $X$ is Hausdorff if for any pair $x, y\in X$, $x\not= y$ there are disjoint open subsets $U, V$ such that $x\in U$ and $y\in V$.
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