Locally connected topological group

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.