# Normal spaces

## 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 $\left\{y\right\}$, then $y$ is in the closure of $\left\{x\right\}$,

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.

Revised on August 5, 2011 20:07:38 by Toby Bartels (64.89.48.241)