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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