The Axioms of Separation are additional conditions to specialize the notion of a topological space. Arbitrary topological spaces are too general an object for many problems of analysis and it is therefore useful to impose extra conditions on a topological space.
|$T_1$||Fréchet Space||For each pair of distinct points $x$ and $y$, their respective neighborhoods $O_x$ and $O_y$ contain one another, i.e. $x \in O_y$ and $y \in O_x$.|
|$T_2$||Hausdorff Space||Every pair of disjoint points in a Hausdorff space has a pair of disjoint neighborhoods.|
- Kolmogorov, A. N., and S. V. Fomin. Introductory Real Analysis. Rev. English ed. New York: Dover Publications, 1975. pg. 78-79.