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
A locally connected topological group is a topological group satisfying the following equivalent conditions:
- The underlying topological space is locally connected.
- 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.