A family of asymmetric Ellis-type theorems

• Autores: S. Andima
• Localización: Houston journal of mathematics, ISSN 0362-1588, Vol. 36, Nº 1, 2010, págs. 181-198
• Idioma: inglés
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• Resumen

  • Bouziad in 1996 generalized theorems of Montgomery (1936) and Ellis (1957), by proving that each C(ech-complete space with a separately continuous group operation must be a topological group. We generalize this result by dropping the requirement that the spaces be Hausdorff or even T1. Our theorems then apply to groups with "asymmetric" topologies, such as the additive group of reals with the upper topology, whose open sets are the open upper rays. We use the fact that each topological space has an associated second topology, which we call the "k-dual", and we consider cases where the bitopological space consisting of the original topology and its k-dual is a "Hausdorff k-bispace", the latter being a bitopological parallel to the topological concept of a Hausdorff k-space, but in which neither topology need be Hausdorff. Suppose a group has a topology in which the group multiplication is separately continuous. Assume also that the bitopological space described above is a Hausdorff k-bitopological space. One of our results is that if the join of the two topologies is C(ech-complete, then inversion is a homeomorphism between the original space and its k-dual, and the group operation is jointly continuous with respect to both topologies. The same conclusion holds more generally if the join is assumed to be a Baire p-space, p-?-fragmentable by a complete sequence of covers.