Difference between revisions of "Separation axioms"
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Revision as of 23:24, 16 March 2010
The separation axioms are a series of definitions in topology that allow the classification of various topological spaces. The following axioms are typically defined: 
. Each axiom is a strictly stronger condition upon the topology than the previous axiom.
Contents
Accessible
In a 
, or an acessible, space, every one-point set is closed.
Hausdorff
![[asy] defaultpen(linewidth(1) + linetype("6 6")); pair A=(0,0),B=(1.6,0),C=(1.6,1),D=(0,1),shiftfactor=(2.5,0.2); /* draw an "open set" using Bezier" */ picture neighborhood(){ picture pic; path p = A{(0.2,-0.6)}..(2*A+B)/3{(0.4,-0.1)}..(A+2*B)/3{(0.4,-0.1)}..B{(0.3,0.7)}..(B+C)/2{(-0.5,0.7)}..C{(-0.1,0.6)}..(2*C+D)/3{(-0.7,0.5)}..(C+2*D)/3{(-0.5,0.5)}..D{(-0.1,-0.5)}..(A+D)/2{(0.6,-0.7)}..cycle; fill(pic,p,rgb(0.9,0.9,0.9)); draw(pic,p); return pic; } /* actual drawing */ add(yscale(1.05)*neighborhood()); add(shift(shiftfactor)*neighborhood()); dot((A+C)/2);dot((A+C)/2 + shiftfactor); /* labels */ label("$x$",(A+C)/2,(1.2,-1.2)); label("$y$",(A+C)/2 + shiftfactor,(1.2,-1.2)); label("$U$",B,(1,-1)); label("$V$",B+shiftfactor,(1,-1)); [/asy]](http://latex.artofproblemsolving.com/3/0/6/3066dd34babf2b102751b8ea3265558553b3ac58.png)
In a 
, or an Hausdorff, space, given any two distinct points 
in a topological space 
, there exists open sets 
such that 
and 
are disjoint.
An example of a space that is but not is the finite complement topology on any infinite space.
Regular
![[asy] defaultpen(linewidth(1) + linetype("6 6")); pair A=(0,0),B=(1.6,0),C=(1.6,1),D=(0,1),shiftfactor=(2.5,0.2); /* draw an "open set" using Bezier" */ picture neighborhood(){ picture pic; path p = A{(0.2,-0.6)}..(2*A+B)/3{(0.4,-0.1)}..(A+2*B)/3{(0.4,-0.1)}..B{(0.3,0.7)}..(B+C)/2{(-0.5,0.7)}..C{(-0.1,0.6)}..(2*C+D)/3{(-0.7,0.5)}..(C+2*D)/3{(-0.5,0.5)}..D{(-0.1,-0.5)}..(A+D)/2{(0.6,-0.7)}..cycle; fill(pic,p,rgb(0.9,0.9,0.9)); draw(pic,p); return pic; } pair SC = (A+C)/2+shiftfactor; path pSC = SC+(-0.25,-0.15)--SC+(-0.25,0.15)--SC+(0.25,0.15)--SC+(0.25,-0.15)--cycle; add(yscale(1.05)*neighborhood()); add(shift(shiftfactor)*neighborhood()); dot((A+C)/2); fill(pSC,rgb(0.7,0.7,0.7)); draw(pSC,linewidth(1)+linetype("6 4")); /* labels */ label("$x$",(A+C)/2,(1.2,-1.2)); label("$B$",(A+C)/2 + shiftfactor,(2,-1)); label("$U$",B,(1,-1)); label("$V$",B+shiftfactor,(1,-1)); [/asy]](http://latex.artofproblemsolving.com/c/5/1/c51c7806453d46709dab7a171272f9fd7e221261.png)
In a 
, or a regular, space, given a point 
and a closed set 
in a topological space that are disjoint, there exists open sets such that 
and are disjoint.
An example of a Hausdorff space that is not regular is the space 
, the k-topology (or in more generality, a subspace of 
consisting of missing a countable number of elements).
Normal
![[asy] defaultpen(linewidth(1) + linetype("6 6")); pair A=(0,0),B=(1.6,0),C=(1.6,1),D=(0,1),shiftfactor=(2.5,0.2); /* draw an "open set" using Bezier" */ picture neighborhood(){ picture pic; path p = A{(0.2,-0.6)}..(2*A+B)/3{(0.4,-0.1)}..(A+2*B)/3{(0.4,-0.1)}..B{(0.3,0.7)}..(B+C)/2{(-0.5,0.7)}..C{(-0.1,0.6)}..(2*C+D)/3{(-0.7,0.5)}..(C+2*D)/3{(-0.5,0.5)}..D{(-0.1,-0.5)}..(A+D)/2{(0.6,-0.7)}..cycle; fill(pic,p,rgb(0.9,0.9,0.9)); draw(pic,p); return pic; } path oSC = (A+C)/2+(-0.25,-0.15)--(A+C)/2+(-0.25,0.15)--(A+C)/2+(0.25,0.15)--(A+C)/2+(0.25,-0.15)--cycle; pair SC = (A+C)/2+shiftfactor; path pSC = SC+(-0.25,-0.15)--SC+(-0.25,0.15)--SC+(0.25,0.15)--SC+(0.25,-0.15)--cycle; add(yscale(1.05)*neighborhood()); add(shift(shiftfactor)*neighborhood()); dot((A+C)/2); fill(pSC,rgb(0.7,0.7,0.7)); draw(pSC,linewidth(1)+linetype("6 4")); fill(oSC,rgb(0.7,0.7,0.7)); draw(oSC,linewidth(1)+linetype("6 4")); label("$A$",(A+C)/2,(2,-1)); label("$B$",(A+C)/2 + shiftfactor,(2,-1)); label("$U$",B,(1,-1)); label("$V$",B+shiftfactor,(1,-1)); [/asy]](http://latex.artofproblemsolving.com/e/2/c/e2c65c0b88dbfee10245d4aad97ae4f05585aa7d.png)
In a 
, or a normal, space, given any two disjoint closed sets 
in a topological space , there exists open sets such that 
and are disjoint.
An example of a normal space that is not regular is the Sorgenfrey plane.
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