A FAMILY OF ASYMMETRIC ELLIS-TYPE THEOREMS

Abstract. Bouziad in 1996 generalized theorems of Montgomery (1936) and Ellis (1957), to prove that every Čech-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 Hausdorff 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 num-ber 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, T k) a Hausdorff k-bispace such that T ∨T k is Čech-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). 1. Introduction an