A topological space is normal if it satisfies:
Often one adds the requirement
(Unlike with regular spaces, is not sufficient here.)
One may also see terminology where a normal space is any space that satsifies , while a -space must satisfy both. This has the benefit that a -space is always also a -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 to be a -space; this convention is also seen.
If instead of , you add only
then the result may be called an -space.
Any space that satisfies both and must be Hausdorff, and every Hausdorff space satisfies , so one may call such a space a normal Hausdorff space; this terminology should be clear to any reader.
Any space that satisfies both and must be regular (in the weaker sense of that term), and every regular space satisfies , so one may call such a space a normal regular space; however, those who interpret ‘normal’ to include usually also interpret ‘regular’ to include , so this term can be ambiguous.
It can be useful to rephrase in terms of only open sets instead of also closed ones:
To spell out the localic case, a normal locale is a frame such that
The Tietze extension theorem applies to normal spaces.
The class of normal spaces was introduced by Tietze (1923) and Aleksandrov–Uryson (1924).