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

Properties

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.

References

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)