Normal Space

Let $X$ be a topological space and $A$ and $B$ are disjoint closed subsets of $X$, then $X$ is said to be normal space if there exist open sets $U$ and $V$ such that $A \subseteq U,{\text{ }}B \subseteq V$, $U \cap V = \phi$.

In other words, a topological space $X$ is said to be a normal space if for any disjoint pair of closed sets $F$ and $G$ there exist open sets $U$ and $V$ such that $F \subseteq U,{\text{ G}} \subseteq V$, $U \cap V = \phi$.

Remarks
• The collection of open sets separating the closed sets is called axiom-$N$.
• It may be noted that some topologists consider the normal space basically ${T_1}$ as well, while others do not.
• Every discrete space contains at least two elements in a normal space.
• Every metric space is a normal space.

T4-Space

A normal ${T_1}$ space is called a ${T_4}$ space.

Theorems
• Every closed subspace of a normal space is a normal space.
• A closed continuous image of a normal space is normal.
• A topological space is normal if and only if for any closed set $A$ and an open set $U$ containing $A$, there is at least one open set $V$ containing $A$ such that $A \subseteq V \subseteq \overline V \subseteq U$.
• Every closed subspace of a ${T_4}$ space is a ${T_4}$ space.
• Every ${T_1}$ and normal space is a regular space.
• If $X$ is a normal space and $f$ is a closed continuous function from $X$ onto a topological space $Y$, then $Y$ is normal as well.