Definition 5.8.1. Let $X$ be a topological space.

1. We say $X$ is irreducible, if $X$ is not empty, and whenever $X = Z_1 \cup Z_2$ with $Z_ i$ closed, we have $X = Z_1$ or $X = Z_2$.

2. We say $Z \subset X$ is an irreducible component of $X$ if $Z$ is a maximal irreducible subset of $X$.

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