Connected implies no proper closed subgroup of finite index

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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

Facts used

  1. Closed subgroup of finite index implies open


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.