General Topology

7 Separation axioms and compactifications

7.1 Separation axioms

We have seen the Hausdorff separation axiom: X is Hausdorff if and only if every pair of distinct points in X can be separated by disjoint neighborhoods. In 1914, before the definitions of topology had become standardized, Felix Hausdorff required what is now known as the Hausdorff separation axiom in his definition of a topology. In their 1935 book Topologie, written in German, Pavel Alexandroff and Heinz Hopf listed the Trennungsaxioms (that is, the separation axioms) which became known as ${\mathit{T}}_0\mathrm{,}{\mathit{T}}_1\mathrm{,}{\mathit{T}}_2\mathrm{,}{\mathit{T}}_3$, and ${\mathit{T}}_4$. They associated the names Kolmogorov, Fréchet, Hausdorff, Vietoris, and Tietze, respectively, with these axioms.

Definition 7.1.1.

Suppose X is a topological space. ...