Topological Groups and Semigroups

1°. A topological group is a group endowed with a Hausdorff topology relative to which the operations of multiplication and inversion are continuous (the latter being therefore a homeomorphism); here the Cartesian product of the group with itself is endowed with the product topology. [The Hausdorff requirement is not included by all authors in the definition of a topological group. It can be in fact relaxed, without restricting the class of groups, to axiom T0: for any two distinct points there is a neighborhood of one of them that does not contain the other.]