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normal (Definition)

A topological space $ X$ is said to be normal if $ X$ is $ T_1$ (i.e. singletons are closed), and for all disjoint closed sets $ D,F\subseteq X$ there exist disjoint open sets $ U$ and $ V$ such that $ C\subseteq U$ and $ D\subseteq V$ (i.e, $ X$ is also $ T_4$).

Some authors do not require the $ T_1$ axiom as part of this definition.



"normal" is owned by Koro. [ full author list (3) | owner history (2) ]
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See Also: separation axioms, Tychonoff space, Hausdorff, completely normal, Hausdorff space, a connected normal space with more than one point is uncountable

Other names:  normality, normal
Keywords:  topology
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Cross-references: axiom, open sets, closed sets, disjoint, closed, singletons, topological space
There are 13 references to this entry.

This is version 10 of normal, born on 2002-01-22, modified 2007-05-23.
Object id is 1532, canonical name is NormalTopologicalSpace.
Accessed 6580 times total.

Classification:
AMS MSC: 54D15 (General topology :: Fairly general properties :: Higher separation axioms )
Pending Errata and Addenda
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Discussion
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normal vs T4 by drini on 2004-10-06 13:04:20

my vote goes with T4 begine weaker than normal, since that's how I was taught in 2 different topo courses at two different unis

after all, both this entry and separation axioms are by Koro, so he should just choose the one he prefers and fix either entry
 f
G -----> H G
p \ /_ ----- ~ f(G)
 \ / f ker f
 G/ker f 
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