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 ${\displaystyle G}$ is a topological space that is also a group such that

${\displaystyle g_{1}:G\times G\to G:(x,y)\mapsto xy}$

is continuous with respect to both ${\displaystyle x}$ and ${\displaystyle y}$. (Note that a topological group is continuous with reference to both variables simultaneously, and ${\displaystyle g_{2}:G\to G:x\mapsto x^{-1}}$ is also required to be continuous. Here ${\displaystyle G\times G}$ is viewed as a topological space with the product topology.)[1]

Clearly, every topological group is a semitopological group. To see that the converse does not hold, consider the real line ${\displaystyle (\mathbb {R} ,+)}$ with its usual structure as an additive abelian group. Apply the lower limit topology to ${\displaystyle \mathbb {R} }$ with topological basis the family ${\displaystyle \{[a,b):-\infty <a<b<\infty \}}$. Then ${\displaystyle g_{1}}$ is continuous, but ${\displaystyle g_{2}}$ is not continuous at 0: ${\displaystyle [0,b)}$ is an open neighbourhood of 0 but there is no neighbourhood of 0 continued in ${\displaystyle g_{2}^{-1}([0,b))}$.

It is known that any locally compact Hausdorff semitopological group is a topological group.[2] Other similar results are also known.[3]

See also

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

References

1. Husain, Taqdir (2018). Introduction to Topological Groups. Courier Dover Publications. p. 27. ISBN 9780486828206.
2. Arhangel’skii, Alexander; Tkachenko, Mikhail (2008). Topological Groups and Related Structures, An Introduction to Topological Algebra. Springer Science & Business Media. p. 114. ISBN 9789491216350.
3. Aull, C. E.; Lowen, R. (2013). Handbook of the History of General Topology. Springer Science & Business Media. p. 1119. ISBN 9789401704700.