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.

• T₀: 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.
• T₁: For any two points in the topological space (S,T) each is an element of an open set which does not contain the other.
• T₂ (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.

• T₃ (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}.
• T₄ (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.
• T₅ (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 T₁ 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 T₁ topological space all finite sets are closed.
• In any T₂ (Hausdorff Space) all compact sets are closed.

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