Difference between revisions of "Separation axioms"

Revision as of 22:23, 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: $T_0 \subset T_1 \subset T_2 \subset T_{2 \frac 12} \subset T_3 \subset T_{3 \frac 12} \subset T_4 \subset T_5 \subset T_6$ . Each axiom is a strictly stronger condition upon the topology than the previous axiom.

Accessible

In a $T_1$ , 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$",B+shiftfactor,(1,-1)); [/asy]$

In a $T_2$ , or an Hausdorff, space, given any two distinct points $x,y$ in a topological space $X$ , there exists open sets $U, V$ such that $x \in U, y \in V$ and $U,V$ are disjoint.

An example of a space that is $T_1$ but not $T_2$ 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$",B+shiftfactor,(1,-1)); [/asy]$

In a $T_3$ , or a regular, space, given a point $x$ and a closed set $A$ in a topological space $X$ that are disjoint, there exists open sets $U,V$ such that $x \in U, A \subset V$ and $U,V$ are disjoint.

An example of a Hausdorff space that is not regular is the space $\mathbb{R}_k$ , the k-topology (or in more generality, a subspace of $\mathbb{R}$ consisting of $\mathbb{R}$ 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$",B+shiftfactor,(1,-1)); [/asy]$

In a $T_4$ , or a normal, space, given any two disjoint closed sets $A,B$ in a topological space $X$ , there exists open sets $U,V$ such that $A \subset U, B \in V$ and $U,V$ are disjoint.

An example of a normal space that is not regular is the Sorgenfrey plane.

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