Connected topological group

Symbol-free definition
A topological group is termed connected if it satisfies the following equivalent conditions:


 * 1) It is connected as a topological space.
 * 2) The connected component of the identity element equals the whole group.

Equivalence of definitions
Definitions (1) and (2) are clearly equivalent.

Alternative definition for a locally connected topological group
For a locally connected topological group, being connected is equivalent to having no proper open subgroup. See locally connected and no proper open subgroup implies connected

In particular, this alternate definition applies to algebraic groups equipped with the Zariski topology, as well as to Lie groups. For more, see equivalence of definitions of connected algebraic group and equivalence of definitions of connected Lie group.

Facts

 * Connected implies no proper open subgroup: The converse holds for locally connected topological groups.