In mathematics, a semitopological group is a topological space with a group action that is continuous with reference 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.
|Algebraic structure → Group theory|
A semitopological group, , is a topological space that is also a group such that
is continuous with reference to both and . (Note that a topological group is continuous with reference to both variables simultaneously, and is also required to be continuous. Here 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 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.
- Husain, Taqdir (2018). Introduction to Topological Groups. Courier Dover Publications. p. 27. ISBN 9780486828206.
- Arhangel’skii, Alexander; Tkachenko, Mikhail (2008). Topological Groups and Related Structures, An Introduction to Topological Algebra. Springer Science & Business Media. p. 114. ISBN 9789491216350.
- Aull, C. E.; Lowen, R. (2013). Handbook of the History of General Topology. Springer Science & Business Media. p. 1119. ISBN 9789401704700.