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 formula_1 is a topological space that is also a group such that
is continuous with respect to both formula_3 and formula_4. (Note that a topological group is continuous with reference to both variables simultaneously, and formula_5 is also required to be continuous. Here formula_6 is viewed as a topological space with the product topology.)
Clearly, every topological group is a semitopological group. To see that the converse does not hold, consider the real line formula_7 with its usual structure as an additive abelian group. Apply the lower limit topology to formula_8 with topological basis the family formula_9. Then formula_10 is continuous, but formula_11 is not continuous at 0: formula_12 is an open neighbourhood of 0 but there is no neighbourhood of 0 continued in formula_13.
It is known that any locally compact Hausdorff semitopological group is a topological group. Other similar results are also known.