Difference between revisions of "Topological separation axioms"
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| + | ==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== | ==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" | {| class="wikitable" border="1" | ||
|- | |- | ||
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|- | |- | ||
! {{M|T_1}} | ! {{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|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}} | ! {{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}} | ! {{M|T_4}} | ||
| + | | A [[normal topological space|normal]] {{M|T_1}} space<ref name="ITTGG"/>. | ||
| + | | [[Every T4 space is a T3 space]] | ||
|} | |} | ||
| + | {{Todo|Picture}} | ||
| + | The {{M|T_i}} notation exists because the German word for "separation axiom" is "Trennungsaxiome"<ref name="ITTGG"/> | ||
| + | ===Recall=== | ||
| + | ====[[Normal topological space]]==== | ||
| + | {{:Normal topological space/Definition}} | ||
| + | ====[[Regular topological space]]==== | ||
| + | {{:Regular 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" | ||
==References== | ==References== | ||
<references/> | <references/> | ||
{{Topology navbox|plain}} | {{Topology navbox|plain}} | ||
{{Definition|Topology}} | {{Definition|Topology}} | ||
Revision as of 23:43, 3 May 2016
Contents
[hide]Motivation
The indiscrete topology and discrete topology are not particularly useful, as first countable and 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 J defined as:
- J:={K}∪{(1,+∞), [0,+∞)} where K is some topology on [0,1]⊂R (power-set / discrete topology being a good example)
We can satisfy any cardinality argument with this (depending on the K) but for anything outside [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, (X,J) is 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 (X,J) is a topological space) Warning:Some authors alter the terms slightly, see the warnings at the bottom
| Space | Meaning | Comment |
|---|---|---|
| T1 | ∀x,y∈X∃U∈J[x≠y⟹(y∈U∧x∉U)][1] | |
| T2 | ∀x,y∈X∃U,V∈J[x≠y⟹(U∩V=∅∧x∈U∧y∈V)][1] | AKA: Hausdorff space. Every T2 space is a T1 space. |
| T3 | A regular T1 space[1]. | Every T3 space is a T2 space |
| T4 | A normal T1 space[1]. | Every T4 space is a T3 space |
TODO: Picture
The Ti notation exists because the German word for "separation axiom" is "Trennungsaxiome"[1]
Recall
Normal topological space
A topological space, (X,J), is said to be normal if[1]:
- ∀E,F∈C(J) ∃U,V∈J[E∩F=∅⟹(U∩V=∅∧E⊆U∧F⊆V)] - (here C(J) denotes the collection of closed sets of the topology, J)
Regular topological space
A topological space, (X,J) is regular if[1]:
- ∀E∈C(J) ∀x∈X−E ∃U,V∈J[U∩V=∅⟹(E⊂U∧x∈V)] - (here C(J) denotes the closed sets of the topology J)
Warning:Note that it is E⊂U not ⊆, the author ([1]) like me is pedantic about this, so it must matter
Warnings
According to[1] some authors define:
- T3 - a regular topological space
- T4 - a normal topological space
This rarely leads to problems as a lot of the topological spaces of interest are Hausdorff spaces (T2) 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"
References
- ↑ Jump up to: 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene
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