T0 topological group

Definition
A topological group is said to be $$T_0$$ if it satisfies the following equivalent conditions:


 * 1) The underlying topological space is $$T_0$$ as a topological space i.e. there is no pair of points such that each is in the closure of the other. See T0 space.
 * 2) The underlying topological space is $$T_1$$ i.e. all points are closed. See T1 space.
 * 3) The underlying topological space is $$T_2$$ or Hausdorff i.e. any two points can be separated by open sets. See Hausdorff space.
 * 4) The underlying topological space is $$T_3$$ or regular i.e. any point and disjoint closed set can be separated by open sets. See regular space.
 * 5) The underlying topological space is completely regular i.e. any point and disjoint closed set can be separated by a continuous function. See completely regular space.

Facts
The equivalence of these multiple definitions follows from the topological group structure, and it is not true for semitopological groups. In other words, being a Hausdorff semitopological group is strictly stronger than being a T0 semitopological group. In fact, for algebraic groups (see algebraic group implies semitopological group, algebraic group not implies topological group), the $$T_0$$ condition is always satisfied, but the Hausdorff condition is not satisfied except in the zero-dimensional case.