Infinite group with cofinite topology is a quasitopological group

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Statement

Suppose G is an infinite group equipped with the cofinite topology, i.e., the proper closed subsets are precisely the finite subsets.

Then, with this topology, G is a quasitopological group: the left multiplication map, right multiplication map, and inverse map are all continuous maps from G to itself.

Note that the corresponding statement is also true for finite groups, but in these cases, the cofinite topology simply becomes the discrete topology, and so the statement is not interesting.

Related facts

Proof

For a set with the cofinite topology, every bijection from the set to itself is a self-homeomorphism. In particular, this means that each of these is a self-homeomorphism:

  • The left multiplication map by any fixed element.
  • The right multiplication map by any fixed element.
  • The inverse map.

Thus, the group is a quasitopological group.