Separation axioms

Summary

The separation axioms are conditions on a topological space that say how well open sets can separate points and closed sets. The main ones, from weak to strong, are T0, T1, Hausdorff, regular and normal.

Remarks

Each of these axioms, together with T1, implies the axioms before it in this list. Each metric space satisfies all of them, by metric spaces are Hausdorff and metrizable spaces are normal. Authors do not agree on all names. For example, some authors include T1 in the definition of a regular or normal space, and some do not.

Explanations

L1 human

The separation axioms measure how many open sets a space has. With more open sets, you can put distinct points, or points and closed sets, into disjoint open sets.