Semitopological group

In mathematics, a semitopological group is a topological space with a group operation 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

g1:G×G→G:(x,y)↦xy

is continuous with respect to both x and y. (Note that a topological group is continuous with reference to both variables simultaneously, and g2:G→G:x↦x−1 is also required to be continuous. Here G×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 (ℝ,+) with its usual structure as an additive abelian group. Apply the lower limit topology to ℝ with topological basis the family {[a,b):−∞<a<b<∞}. Then g1 is continuous, but g2 is not continuous at 0: [0,b) is an open neighbourhood of 0 but there is no neighbourhood of 0 contained in g2−1([0,b)).

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

See also

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.