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 .

• 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 other do not.
• Every discrete space containing at least two elements in a normal space.
• Every metric space is a normal space.


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

• Every closed subspace of normal space is normal space.
• A closed continuous image of a normal space is normal.
• A topological space is normal if and only if 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 closed continuous function from X onto a topological space Y, then Y is normal as well.