In topology and related branches of mathematics, a normal space is a topological space in which any two disjoint closed sets have disjoint open neighborhoods. Such spaces need not be Hausdorff in general. A normal Hausdorff space is called a T₄ space. Strengthenings of these concepts are detailed in the article below and include completely normal spaces and perfectly normal spaces, and their Hausdorff variants: T₅ spaces and T₆ spaces. All these conditions are examples of separation axioms.

A topological space X is a normal space if, given any disjoint closed sets E and F, there are 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 hereditarily normal space, is a topological space X such that every subspace of X 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. Also, X is completely normal if and only if every open subset of X is normal with the subspace topology.

A T₅ space, or completely T₄ space, is a completely normal T₁ space X, which implies that X is Hausdorff; equivalently, every subspace of X must be a T₄ space.

A perfectly normal space is a topological space ${\displaystyle X}$ in which every two disjoint closed sets ${\displaystyle E}$ and ${\displaystyle F}$ can be precisely separated by a function, in the sense that there is a continuous function ${\displaystyle f}$ from ${\displaystyle X}$ to the interval ${\displaystyle [0,1]}$ such that ${\displaystyle f^{-1}(\{0\})=E}$ and ${\displaystyle f^{-1}(\{1\})=F}$. This is a stronger separation property than normality, as by Urysohn's lemma disjoint closed sets in a normal space can be separated by a function, in the sense of ${\displaystyle E\subseteq f^{-1}(\{0\})}$ and ${\displaystyle F\subseteq f^{-1}(\{1\})}$, but not precisely separated in general. It turns out that X is perfectly normal if and only if X is normal and every closed set is a G_δ set. Equivalently, X is perfectly normal if and only if every closed set is the zero set of a continuous function. The equivalence between these three characterizations is called Vedenissoff's theorem. Every perfectly normal space is completely normal, because perfect normality is a hereditary property.

A T₆ space, or perfectly T₄ space, is a perfectly normal Hausdorff space.

Note that the terms "normal space" and "T₄" and derived concepts occasionally have a different meaning. (Nonetheless, "T₅" always means the same as "completely T₄", whatever the meaning of T₄ may be.) The definitions given here are the ones usually used today. For more on this issue, see History of the separation axioms.