Normal topological space

Definition

A topological space, $(X,\mathcal{ J })$, is said to be normal if:

• $\forall E,F\in C(\mathcal{J})\ \exists U,V\in\mathcal{J}[E\cap F=\emptyset\implies(U\cap V=\emptyset\wedge E\subseteq U\wedge F\subseteq V)]$ - (here $C(\mathcal{J})$ denotes the collection of closed sets of the topology, $\mathcal{J}$)

Equivalent statements

• A topological space is normal if and only if for each closed set, E, and each open set, W, containing E there exists an open set U containing E such that the closure of U is strictly a subset of W

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See also

• Topological separation axioms
• Regular topological space
• Urysohn's lemma
• Tietze extension theorem