San José State University
Thayer Watkins
Silicon Valley

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 Ti-axioms.

Disjoint sets means that their closures are separate.

Some Theorems That Apply for the Various Ti Topological Spaces