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The Stacks project

Lemma 5.11.5. Let X be a topological space. The following are equivalent:

  1. X is catenary,

  2. X has an open covering by catenary spaces.

Moreover, in this case any locally closed subspace of X is catenary.

Proof. Suppose that X is catenary and that U \subset X is an open subset. The rule T \mapsto \overline{T} defines a bijective inclusion preserving map between the closed irreducible subsets of U and the closed irreducible subsets of X which meet U. Using this the lemma easily follows. Details omitted. \square


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