Let $(X,\tau)$ be a [normal topological space](/page/Normal%20Topological%20Space). Suppose that $(X,\tau)$ is a $T_1$ [topological space](/page/Topological%20Space). Then $(X,\tau)$ is Hausdorff: for every pair of distinct points $x,y \in X$, there exist open sets $U,V \in \tau$ such that
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\begin{align*}
x \in U, \quad y \in V, \quad U \cap V = \varnothing.
\end{align*}