Definition 5.8.1 (004V)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 5: Topology
Section 5.8: Irreducible components
Definition 5.8.1 (cite)

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

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 $X = Z_2$ .
We say $Z \subset X$ is an irreducible component of $X$ if $Z$ is a maximal irreducible subset of $X$ .

Comments (2)

Comment #8722 by Roy Shtoyerman on November 09, 2023 at 20:02

It might be lacking a definition of an irreducible subset . I assume it's just irreducible as a space w.r.t. the subset topology, but it might be useful to elaborate for clarity.

Comment #9353 by Stacks project on June 04, 2024 at 10:32

We always use the induced topology on subsets, see Section 5.6. Going to leave as is.

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