Exercise 111.6.20. Prove that a Noetherian topological space $X$ has only finitely many irreducible components, say $X_1, \ldots , X_ n$, and that $X = X_1\cup X_2\cup \ldots \cup X_ n$. (Note that any $X$ is always the union of its irreducible components, but that if $X = {\mathbf R}$ with its usual topology for instance then the irreducible components of $X$ are the one point subsets. This is not terribly interesting.)

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