nLab
normal space

Idea

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

Definitions

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 $U\wedge V=\perp$.

Remarks

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

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).

The Tietze extension theorem applies to normal spaces.

References

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