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Homenormal

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# normal

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.

Keywords:

topology

Related:

SeparationAxioms, Tychonoff, Hausdorff, CompletelyNormal, T2Space, AConnectedNormalSpaceWithMoreThanOnePointIsUncountable2, AConnectedNormalSpaceWithMoreThanOnePointIsUncountable, ApplicationsOfUrysohnsLemmaToLocallyCompactHausdorffSpaces

Synonym:

normality, normal

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

54D15*no label found*

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## Corrections

page images mode by alozano ✓

a property by matte ✓

synonyms by matte ✓

T4 vs. normal by matte ✓

inconsistency in definitions by Mathprof ✓

a property by matte ✓

synonyms by matte ✓

T4 vs. normal by matte ✓

inconsistency in definitions by Mathprof ✓

## Comments

## normal vs T4

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