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: Whenever (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). This generalizes the classical Ellis theorem, because T = T~k when (X, T) is locally compact Hausdorff.
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