Normal space

In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T₄: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T₄ space. These conditions are examples of separation axioms and their further strengthenings define completely normal Hausdorff spaces, or T₅ spaces, and perfectly normal Hausdorff spaces, or T₆ spaces.

Definitions

A topological space X is a normal space if, given any disjoint closed sets E and F, there are open neighbourhoods U of E and V of F that are also disjoint. More intuitively, this condition says that E and F can be separated by neighbourhoods.

A T₄ space is a T₁ space X that is normal; this is equivalent to X being normal and Hausdorff.

A completely normal space or a hereditarily normal space is a topological space X such that every subspace of X with subspace topology is a normal space. It turns out that X is completely normal if and only if every two separated sets can be separated by neighbourhoods.