Semitopological group


In mathematics, a semitopological group is a topological space with a group action that is continuous with respect to each variable considered separately. It is a weakening of the concept of a topological group; all topological groups are semitopological groups but the converse does not hold.

Formal definition

A semitopological group is a topological space that is also a group such that
is continuous with respect to both and.
Clearly, every topological group is a semitopological group. To see that the converse does not hold, consider the real line with its usual structure as an additive abelian group. Apply the semiopen topology to with topological basis the family. Then is continuous, but is not continuous at 0: is an open neighbourhood of 0 but there is no neighbourhood of 0 continued in.
It is known that any locally compact Hausdorff semitopological group is a topological group. Other similar results are also known.