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The Hausdorff Axiom
and Other Separation Axioms for Topological Spaces

The fully general, abstract topological spaces (S,T), where S is
a set and T is a collection of subsets of S that is closed under
arbitrary unions and finite intersections and includes both the null set
∅ and the whole set S, is of limited interest. It is of
interest mainly in reducing concepts such as compactness to their
barest bones form. In order to get more structure and hence more
interesting theorems Alexandroff and Hopf formulated a sequence of
successively more restrictive axioms called in English separation
axioms but known in German as trennungsaxioms and hence
designated as T_{i}-axioms.

T_{0}: For any two points in the topological space (S,T) at
least one if an element of an open set which does not contain the other.

T_{1}: For any two points in the topological space (S,T)
each is an element of an open set which does not contain the other.

T_{2} (Hausdorff Space Axiom): For any two points p and q
in the topological space (S,T)
there is a pair of disjoint open sets, one containing {p} and not containgin
{q} and the other
containing {q} and not {p}.

Disjoint sets means that their closures
are separate.

T_{3} (Regular Space Axiom): For any closed set C
in the topological space (S,T) and any point p
not in C
there is a pair of disjoint open sets, one containing C
and one containing {p}.

T_{4} (Normal Space Axiom): For any pair of disjoint closed
sets G and H
in the topological space (S,T)
there is a pair of disjoint open sets, one containing G
and the other containing H.

T_{5} (Completely Normal Space Axioms): In a topological space (S,T)
1, For any two points p and q in S there exist a pair of open sets such
that one contains {p} and not {q} and the other contains {q} and not {p}
(the T_{1} axiom).
For any pair of
sets G and H such that neither have any points in common with the closure
of the other
there is a pair of disjoint open sets, one containing G
and the other containing H.

Some Theorems That Apply for the Various T_{i} Topological
Spaces

In any T_{1} topological space all finite sets are closed.

In any T_{2} (Hausdorff Space) all compact sets are closed.