normal space

A normal space is a space (typically a topological space) which satsifies one of the stronger separation axioms.

A topological space $X$ is **normal** if it satisfies:

- $T_4$: for every two closed disjoint subsets $A,B \subset X$ there are (optionally open) neighborhoods $U \supset A$, $V \supset B$ such that $U \cap V$ is empty.

Often one adds the requirement

- $T_1$: every point in $X$ is closed.

(Unlike with regular spaces, $T_0$ is not sufficient here.)

One may also see terminology where a **normal space** is any space that satsifies $T_4$, while a **$T_4$-space** must satisfy both. This has the benefit that a $T_4$-space is always also a $T_3$-space while still having a term available for the weaker notion. On the other hand, the reverse might make more sense, since you would expect any space that satisfies $T_4$ to be a $T_4$-space; this convention is also seen.

If instead of $T_1$, you add only

- $R_0$: if $x$ is in the closure of $\{y\}$, then $y$ is in the closure of $\{x\}$,

then the result may be called an **$R_3$-space**.

Any space that satisfies both $T_4$ and $T_1$ must be Hausdorff, and every Hausdorff space satisfies $T_1$, so one may call such a space a **normal Hausdorff space**; this terminology should be clear to any reader.

Any space that satisfies both $T_4$ and $R_0$ must be regular (in the weaker sense of that term), and every regular space satisfies $R_0$, so one may call such a space a **normal regular space**; however, those who interpret ‘normal’ to include $T_1$ usually also interpret ‘regular’ to include $T_1$, so this term can be ambiguous.

It can be useful to rephrase $T_4$ in terms of only open sets instead of also closed ones:

- $T_4$: if $G,H \subset X$ are open and $G \cup H = X$, then there exist open sets $U,V$ such that $U \cup G$ and $V \cup H$ are still $X$ but $U \cap V$ is empty.

This definition is suitable for generalisation to locales and also for use in constructive mathematics (where it is not equivalent to the usual one).

To spell out the localic case, a **normal locale** is a frame $L$ such that

- $T_4$: if $G,H \in L$ are opens and $G \vee H = \top$, then there exist opens $U,V$ such that $U \vee G$ and $V \vee H$ are still $\top$ but $U \wedge V = \bot$.

Every metric space is normal Hausdorff. Every normal Hausdorff space is an Urysohn space, a fortiori regular and a fortiori Hausdorff.

Every regular second countable space is normal. Every paracompact Hausdorff space is normal (Dieudonné’s theorem). See at compact Hausdorff spaces are normal

The Tietze extension theorem applies to normal spaces.

The class of normal spaces was introduced by Tietze (1923) and Aleksandrov–Uryson (1924).

- Ryszard Engelking,
**General topology**, (Monographie Matematyczne, tom 60) Warszawa 1977; expanded Russian edition Mir 1986.

Revised on April 16, 2017 08:31:23
by Urs Schreiber
(92.218.150.85)