Locally connected and no proper open subgroup implies connected

Statement

Suppose $G$ is a topological group that is locally connected and has no proper open subgroup. Then, $G$ is a connected topological group.

Facts used

1. Locally connected implies connected component of identity is open subgroup

Related facts

• Connected implies no proper open subgroup
• No proper open subgroup not implies connected

Proof

The proof follows directly from Fact (1).