A family of asymmetric Ellis-type theorems

Bouziad in 1996 generalized theorems of Montgomery (1936) and Ellis (1957), to prove that every Cech complete space with a separately continuous group operation must be a topological group. We generalize these results in a new direction, by dropping the requirement that the spaces be T2 or even T1. Our theorems then become applicable to groups with asymmetric topologies, such as the group of real numbers with the upper topology, whose open sets are the open upper rays. We first show a generic Ellis-type theorem for groups with a Hausdor k-bitopological structure whose symmetrization belongs to a class of k-spaces for which a classical Ellis-type theorem is known. We then develop a number of specific cases, including the following: Let (G T) be a group with a topology making multiplication separately continuous, whose k dual T k makes (G T Tk) a Hausdor k-bispace such that T T k is Cech complete. Then multiplication is jointly continuous with respect to both T and T k, and inversion is a homeomorphism between (G T) and (G T k)