Locally connected topological group

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 locally connected topological group is a topological group satisfying the following equivalent conditions:

1. The underlying topological space is locally connected.
2. Every open subset containing the identity element contains an open subset, also containing the identity element, whose closure is connected.

Equivalence of definitions

(1) is a priori stronger than (2) because (2) is just (1) applied to the identity element. However, by the fact that any toopological group is a homogeneous space, we conclude that the two definitions are equivalent.