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 $$G$$ is a topological space that is also a group such that


 * $$g_1: G \times G \to G : (x,y)\mapsto xy$$

is continuous with respect to both $$x$$ and $$y$$. (Note that a topological group is continuous with reference to both variables simultaneously, and $$g_2: G\to G : x \mapsto x^{-1}$$ is also required to be continuous. Here $$ G \times G $$ 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 $$(\mathbb{R},+)$$ with its usual structure as an additive abelian group. Apply the lower limit topology to $$\mathbb{R}$$ with topological basis the family $$\{[a,b):-\infty < a < b < \infty \}$$. Then $$g_1$$ is continuous, but $$g_2$$ is not continuous at 0: $$[0,b)$$ is an open neighbourhood of 0 but there is no neighbourhood of 0 continued in $$g_2^{-1}([0,b))$$.

It is known that any locally compact Hausdorff semitopological group is a topological group. Other similar results are also known.