View source for Topological separation axioms

__TOC__
==Motivation==
The [[indiscrete topology]] and [[discrete topology]] are not particularly useful, as [[first countable topological space|first countable]] and [[second countable topological space|second countable]] ''[[topological spaces]]'' show by trying to restrict the cardinality of the [[topology]] we may derive some interesting properties that are not true for [[topological spaces]] in general. [[Cardinality]] arguments are "weak" in the sense of they don't really provide a useful constraint on the topology. Consider for example the topology {{M|\mathcal{J} }} defined as:
* {{M|1=\mathcal{J}:=\{\mathcal{K}\}\cup\{ (1,+\infty),\ [0,+\infty)\} }} where {{M|\mathcal{K} }} is some topology on {{M|[0,1]\subset\mathbb{R} }} (power-set / discrete topology being a good example) 
We can satisfy any cardinality argument with this (depending on the  {{M|\mathcal{K} }}) but for anything outside {{M|[0,1]}} we have no information. This is what I mean by cardinality arguments are weak. They don't govern the space. 

However, suppose that a [[topological space]], {{Top.|X|J}} is [[Hausdorff space|Hausdorff]] say, now we have a "strong" property (or requirement for the space), because it applies to ALL points equally well.
{{Todo|Flesh out}}
==Overview==
(Here {{Top.|X|J}} is a [[topological space]])
{{Warning|Some authors alter the terms slightly, see the warnings at the bottom}}
{| class="wikitable" border="1"
|-
! Space
! Meaning
! Comment
|-
! {{M|T_1}}
| {{M|1=\forall x,y\in X\exists U\in\mathcal{J}[x\ne y\implies (y\in U\wedge x\notin U)]}}{{rITTGG}}
|
|-
! {{M|T_2}}
| {{M|1=\forall x,y\in X\exists U,V\in\mathcal{J}[x\ne y\implies(U\cap V=\emptyset\wedge x\in U\wedge y\in V)]}}<ref name="ITTGG"/>
| {{AKA}}: [[Hausdorff space]]. [[Every T2 space is a T1 space]].
|-
! {{M|T_3}}
| A [[regular topological space|regular]] {{M|T_1}} space<ref name="ITTGG"/>.
| [[Every T3 space is a T2 space]]
|-
! {{M|T_4}}
| A [[normal topological space|normal]] {{M|T_1}} space<ref name="ITTGG"/>.
| [[Every T4 space is a T3 space]], [[All metric spaces are T4 spaces]]<ref group="Note">Show all [[metric space|metric spaces]] are [[Hausdorff space|Hausdorff spaces]] is incredibly easy. However there is a stronger result, [[All metric spaces are T4 spaces]], Then as [[every T4 space is a T3 space]] and [[every T3 space is a T2 space]] and a [[T2 space]] is the same thing as a [[Hausdorff space]] we can prove it this way instead. </ref>
|}
{{Todo|Picture}}
The {{M|T_i}} notation exists because the German word for "separation axiom" is "Trennungsaxiome"<ref name="ITTGG"/>
===Recall===
====[[Regular topological space]]====
{{:Regular topological space/Definition}}
====[[Normal topological space]]====
{{:Normal topological space/Definition}}

==Warnings==
According to<ref name="ITTGG"/> some authors define:
* {{M|T_3}} - a [[regular topological space]]
* {{M|T_4}} - a [[normal topological space]]
This rarely leads to problems as a lot of the [[topological spaces]] of interest are [[Hausdorff spaces]] ({{M|T_2}}) and on such spaces the definitions coincide. 

We shall use the definitions as given above, because we already have a term for "normal topological spaces" - that term is "normal topological space"
==See also==
* [[Urysohn's lemma]] - for ''[[disjoint]]'' [[closed set|closed sets]] of a [[normal topological space]], {{Top.|X|J}}, there exists a [[continuous function]], {{M|f:X\rightarrow [0,1]\subset\mathbb{R} }} such that {{M|f}} is {{M|0}} on the first closed set and {{M|f}} is {{M|1}} on the other.
==Notes==
<references group="Note"/>
==References==
<references/>
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