T0 topological group

From Groupprops
Jump to: navigation, search
This article defines a property that can be evaluated for a topological group (usually, a T0 topological group)
View a complete list of such properties

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.

Equivalence of definitions

Further information: equivalence of definitions of T0 topological group

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.