An asymmetric Ellis theorem

Abstract: In 1957 Robert Ellis proved that a group with a locally compact Hausdorff topology making all translations continuous also has jointly continuous multiplication and continuous inversion, and is thus a topological group. The theorem does not apply to locally compact asymmetric spaces such as the reals with addition and the topology of upper open rays. We first show a bitopological Ellis theorem, and then introduce a generalization of locally compact Hausdorff, called locally skew compact, and a topological dual, , to obtain the following asymmetric Ellis theorem which applies to the example above: Whenever is a group with a locally skew compact topology making all translations continuous, then multiplication is jointly continuous in both and , and inversion is a homeomorphism between and . This generalizes the classical Ellis theorem, because when is locally compact Hausdorff. [Copyright &y& Elsevier]