Separation Axioms (Hausdorff & beyond)

The T_0 through T_4 hierarchy: points, closed sets, and disjoint open separation.

The idea

The separation axioms grade topological spaces by how well disjoint open sets can pull objects apart. The topology axioms alone permit spaces in which no open set distinguishes two distinct points, and spaces in which one sequence converges to several limits at once; the separation axioms are the extra hypotheses that rule such spaces out, arranged as a scale so that a theorem can assume exactly as much as its proof uses.

Definition (Separation axioms).

A space is $T_1$ if for any two distinct points, each has an open set that misses the other. It is $T_2$, or Hausdorff, if any two distinct points $x$ and $y$ have disjoint open neighbourhoods $U$ and $V$. It is $T_3$, or regular, if it is $T_1$ and any point $x$ and closed set $A$ not containing $x$ have disjoint open neighbourhoods. It is $T_4$, or normal, if it is $T_1$ and any two disjoint closed sets $A$ and $B$ have disjoint open neighbourhoods.

Each level asks the same question of a larger pair of objects: can these two be placed in disjoint open sets? The $T_1$ condition is equivalent to every one-point set being closed. At the Hausdorff level limits become unique, which is why virtually every space met in practice is assumed Hausdorff. Because $T_3$ and $T_4$ are stated together with $T_1$, points themselves count as closed sets and each level implies the ones below.

Ways to work on it

• Walkthrough. Read the T_1 through T_4 axioms and the implications between them.
• Practice. Classify a space by the highest separation axiom it satisfies.
• Hardest. The Sorgenfrey plane: regular but not normal.

Not sure where to start? Take the ten-question placement test.