Semitopological group

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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.

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Formal definition

A semitopological group is a topological space that is also a group such that

is continuous with respect 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.) [1]

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 lower limit 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. [2] Other similar results are also known. [3]

See also

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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.