An asymmetric Ellis theorem

In 1957 Robert Ellis proved that a group with a locally compact Hausdorff topology T 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, T k , to obtain the following asymmetric Ellis theorem which applies to the example above: er ( X , ⋅ , T ) is a group with a locally skew compact topology making all translations continuous, then multiplication is jointly continuous in both ( X , ⋅ , T ) and ( X , ⋅ , T k ) , and inversion is a homeomorphism between ( X , T ) and ( X , T k ) . eneralizes the classical Ellis theorem, because T = T k when ( X , T ) is locally compact Hausdorff.