Semitopological group

In mathematics, a semitopological group is a topological space with a group action that is continuous with reference to a 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 reference 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.)^[1]

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 semiopen 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 contined in $g_{2}^{-1}([0,b))$ .

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

References

↑ 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.

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[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.

Semitopological group

Formal definition

See also

References