Lemma 5.11.5. Let X be a topological space. The following are equivalent:
X is catenary,
X has an open covering by catenary spaces.
Moreover, in this case any locally closed subspace of X is catenary.
Lemma 5.11.5. Let X be a topological space. The following are equivalent:
X is catenary,
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
Comments (0)
There are also: