modified from https://en.m.wikipedia.org/wiki/Separation_axiom Sep 3 2016.
by Misha Gavrilovich Sep 8 2016, Apr 2017 http://mishap.sdf.org/wiki/Separation_axioms.txt
Separation Axioms.
In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes called Tychonoff separation axioms, after Andrey Tychonoff.
The separation axioms are axioms only in the sense that, when defining the notion of topological space, one could add these conditions as extra axioms to get a more restricted notion of what a topological space is. The modern approach is to fix once and for all the axiomatization of topological space and then speak of kinds of topological spaces. However, the term "separation axiom" has stuck. The separation axioms are denoted with the letter "T" after the German Trennungsaxiom, which means "separation axiom."
Let X be a topological space. Then two points x and y in X are __topologically distinguishable__
iff the map {x<->y} --> X is not continuous, i.e.
iff %if they do not have exactly the same neighbourhoods (or equivalently the same open neighbourhoods);
at least one of them has an open neighbourhood which is not a neighbourhood of the other.
Two points x and y are __separated__ iff neither {x->y} --> X nor {x->y} --> X is continuous,
i.e~each of them has a neighbourhood that is not a neighbourhood of the other;
in other words, neither belongs to the other's closure, x not in cl x and y not in cl x .
More generally, two subsets A and B of X are __separated__ iff each is disjoint from the other's closure,
i.e.~ A /\ cl B = B /\ cl A = {} .
(The closures themselves do not have to be disjoint.) In other words, the map
i_AB : X --> {A <-> x <-> B}
sending the subset A to the point A , the subset B to the point B , and the rest to the point x ,
factors both as
X --> {A <-> U_A -> x <-> B} --> {A=U_A <-> x <-> B}
and
X --> {A <-> x <- U_B <-> B} --> {A <-> x <-> U_B=B}
here the preimage of x,B , resp. x,A is a closed subset containing B , resp. A , and disjoint from A , resp. B .
All of the remaining conditions for separation of sets may also be applied to points (or to a point and a set)
by using singleton sets. Points x and y will be considered separated, by neighbourhoods,
by closed neighbourhoods, by a continuous function, precisely by a function,
iff their singleton sets {x} and {y} are separated according to the corresponding criterion.
Subsets A and B are __separated by neighbourhoods__ iff
A and B have disjoint neighbourhoods, i.e.
iff
i_AB : X --> {A <-> x <-> B} factors as
X --> {A <-> U_A -> x <- U_B <-> B} --> {A=U_A <-> x <-> U_B=B}
here the disjoint neighbourhoods of A and B are the preimages of open subsets {A,U_A} and {U_B,B} of
{A <-> U_A -> x <- U_B <-> B} , resp.
They are __separated by closed neighbourhoods__
iff they have disjoint closed neighbourhoods, i.e.
i_AB factors as
X --> {A <-> U_A -> U'_A <- x -> U'_B <- U_B <-> B} --> {A<->U_A=U'_A= x = U'_B=U_B<->B}
.
They are __separated by a continuous function__ iff
there exists a continuous function f from the space X to the real line R such that f(A)=0 and f(B)=1 ,
i.e.
the map i_AB factors as
X --> {0'} \union [0,1] \union {1'} --> {A <-> x <-> B}
where points 0',0 and 1,1' are topologically indistinguishable,
and 0' maps to A , and 1' maps to B , and [0,1] maps to x .
Finally, they are __precisely separated by a continuous function__
iff there exists a continuous function f from X to R such that the preimage f^{ - 1}({0})= A and f^{ - 1}({1})=B .
i.e.~iff i_AB factors as
X --> [0,1] --> {A <-> x <-> B}
where 0 goes to point A and 1 goes to point B .
These conditions are given in order of increasing strength:
Any two topologically distinguishable points must be distinct, and any two separated points must be topologically distinguishable. Any two separated sets must be disjoint, any two sets separated by neighbourhoods must be separated, and so on.
%For more on these conditions (including their use outside the separation axioms), see the articles Separated sets and Topological distinguishability.
%Main definitions
The definitions below all use essentially the preliminary definitions above.
In all of the following definitions, X is again a topological space.
+ X is T0, or Kolmogorov, if any two distinct points in X are topologically
distinguishable. (It will be a common theme among the separation axioms to have
one version of an axiom that requires T0 and one version that doesn't.)
As a formula, this is expressed as
{x<->y} ---> {x=y} /_ X --> {*}
+ X is R0, or symmetric, if any two topologically distinguishable points in X
are separated, i.e.
{x->y} --> {x<->y} /_ X --> {*}
+ X is T1, or accessible or Frechet, if any two distinct points in X are
separated, i.e.
{x->y} ---> {x=y} /_ X ---> {*}
Thus, X is T1 if and only if it is both T0 and R0. (Although you may
say such things as "T1 space", "Frechet topology", and "Suppose that the
topological space X is Frechet", avoid saying "Frechet space" in this
context, since there is another entirely different notion of Frechet space in
functional analysis.)
+ X is R1, or preregular, if any two topologically distinguishable points in
X are separated by neighbourhoods. Every R1 space is also R0.
+ X is weak Hausdorff, if the image of every continuous map from a compact
Hausdorff space into X is closed. All weak Hausdorff spaces are T1, and all
Hausdorff spaces are weak Hausdorff.
+ X is Hausdorff, or T2 or separated, if any two distinct points in X are
separated by neighbourhoods, i.e.
{x,y} (--> X /_ {x->X<-y} --> {x=X=y}
Thus, X is Hausdorff if and only if it is both T0
and R1. Every Hausdorff space is also T1.
+ X is T2\frac12 , or Urysohn, if any two distinct points in X are separated by
closed neighbourhoods, i.e.
{x,y} >--> X /_ {x->x'<-X->y'<-y} --> {x=x'=X=y'=y}
Every T2 \frac12 space is also Hausdorff.
+ X is completely Hausdorff, or completely T2, if any two distinct points in
X are separated by a continuous function, i.e.
{x,y} (--> X /_ [0,1]-->{*}
where {x,y} >--> X runs through all injective maps from the discrete two
point space {x,y} .
Every completely Hausdorff space is
also T2 \frac 12 ½ .
+ X is regular if, given any point x and closed subset F in X such that x does
not belong to F , they are separated by neighbourhoods, i.e.
{x} ---> X /_ {x->X<-U->F} ---> {x=X=U->F}
(In fact, in a regular
space, any such x and F will also be separated by closed neighbourhoods.) Every
regular space is also R1.
+ X is regular Hausdorff, or T3, if it is both T0 and regular.[1] Every
regular Hausdorff space is also T2\frac12 .
+ X is completely regular if, given any point x and closed set F in X such
that x does not belong to F , they are separated by a continuous function, i.e.
{x} ---> X /_ [0,1] \cup {F} ---> {x->F}
where points F and 1 are topologically indistinguishable, [0,1] goes to x ,
and F goes to F .
Every
completely regular space is also regular.
+ X is Tychonoff, or T3 \frac12 , completely T3, or completely regular Hausdorff, if
it is both T0 and completely regular.[2] Every Tychonoff space is both regular
Hausdorff and completely Hausdorff.
+ X is normal if any two disjoint closed subsets of X are separated by
neighbourhoods, i.e.
{} -->X /_ {x<-x'->X<-y'->y} --> {x<-x'=X=y'->y}
In fact, by Urysohn lemma a space is normal if and only if any two disjoint
closed sets can be separated by a continuous function, i.e.
{} --> X /_ {0'} \cup [0,1] \cup {1'} --> {0=0'->x<-1=1'}
where points
0',0 and 1,1' are topologically indistinguishable,
[0,1] goes to x , and both 0,0' map to point 0=0' , and both 1,1' map to point
1=1' .
+ X is normal Hausdorff, or T4, if it is both T1 and normal. Every normal
Hausdorff space is both Tychonoff and normal regular.
+ X is completely normal if any two separated sets A and B are separated by
neighbourhoods U=)A and V=)B
such that U and V do not intersect, i.e.????
{} ---> X /_ {X<-A<->U->U'<-W->V'<-V<->B->X} ---> {U=U',V'=V}
Every completely normal space is also normal.
%
% {} --> X /_ \\ {X<-x<->U->U'<-W->V'<-V<->y->X} -->
%{X<-x<->U=U'<-W->V'=V<->y->X}
% or in short
+ X is perfectly normal if any two disjoint closed sets are precisely
separated by a continuous function, i.e.
{}-->X /_ [0,1]-->{0<-X->1}
where (0,1) goes to the open point X , and 0 goes to 0 , and 1 goes to 1 .
Every perfectly normal space is also
completely normal.