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 [math]\displaystyle{ G }[/math] is a topological space that is also a group such that

[math]\displaystyle{ g_1: G \times G \to G : (x,y)\mapsto xy }[/math]

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

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

See also

• Lie group
• Algebraic group
• Compact group
• Topological ring