Quasitopological group not implies topological group

Statement

It is possible to have a quasitopological group that is not a topological group.

Related facts

• Topological group implies quasitopological group
• Algebraic group implies quasitopological group
• Algebraic group not implies topological group

Facts used

1. Infinite group with cofinite topology is a quasitopological group
2. Infinite group with cofinite topology is not a topological group

Proof

The proof follows directly form Facts (1) and (2) and the observation that infinite groups exist.