Connected implies no proper closed subgroup of finite index

Statement

In a connected topological group, there cannot be any proper closed subgroup of finite index (i.e., a closed subgroup that is also a subgroup of finite index).

Related facts

• Connected implies no proper open subgroup
• Open subgroup implies closed
• Closed subgroup of finite index implies open
• Compact implies every open subgroup has finite index

Facts used

1. Closed subgroup of finite index implies open

Proof

The proof is direct from Fact (1), and the observation that the existence of a proper subgroup (and hence a proper nonempty subset) that is both closed and open means that the group is not connected.