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 topological group is termed connected if it satisfies the following equivalent conditions:
- It is connected as a topological space.
- It has no proper open subgroup
- The connected component of the identity element equals the whole group.
Equivalence of definitions
Definitions (1) and (3) are clearly equivalent. For proof of (1) implies (2), use connected implies no proper open subgroup. The reverse implication is somewhat trickier.