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, ÃÂÃÂÃÂÃÂ·,T ) is a group with a locally skew compact topology making all translations continuous, then multiplication
is jointly continuous in both (X, ÃÂÃÂÃÂÃÂ·,T ) and (X, ÃÂÃÂÃÂÃÂ·,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.