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.