nLab
normal space

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 XX is normal if it satisfies:

Often one adds the requirement

  • T 1T_1: every point in XX is closed.

(Unlike with regular spaces, T 0T_0 is not sufficient here.)

One may also see terminology where a normal space is any space that satsifies T 4T_4, while a T 4T_4-space must satisfy both. This has the benefit that a T 4T_4-space is always also a T 3T_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 4T_4 to be a T 4T_4-space; this convention is also seen.

If instead of T 1T_1, you add only

  • R 0R_0: if xx is in the closure of {y}\{y\}, then yy is in the closure of {x}\{x\},

then the result may be called an R 3R_3-space.

Any space that satisfies both T 4T_4 and T 1T_1 must be Hausdorff, and every Hausdorff space satisfies T 1T_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 4T_4 and R 0R_0 must be regular (in the weaker sense of that term), and every regular space satisfies R 0R_0, so one may call such a space a normal regular space; however, those who interpret ‘normal’ to include T 1T_1 usually also interpret ‘regular’ to include T 1T_1, so this term can be ambiguous.

It can be useful to rephrase T 4T_4 in terms of only open sets instead of also closed ones:

  • T 4T_4: if G,HXG,H \subset X are open and GH=XG \cup H = X, then there exist open sets U,VU,V such that UGU \cup G and VHV \cup H are still XX but UVU \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 LL such that

  • T 4T_4: if G,HLG,H \in L are opens and GH=G \vee H = \top, then there exist opens U,VU,V such that UGU \vee G and VHV \vee H are still \top but UV=U \wedge V = \bot.

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)