Example 5.8.13. Recall that a topological space $X$ is Hausdorff iff for every distinct pair of points $x, y \in X$ there exist disjoint opens $U, V \subset X$ such that $x \in U$, $y \in V$. In this case $X$ is irreducible if and only if $X$ is a singleton. Similarly, any subset of $X$ is irreducible if and only if it is a singleton. Hence a Hausdorff space is sober.
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