On uncountable hypersimple unidimensional theories

We extend the dichotomy between 1-basedness and supersimplicity proved in Shami (J Lond Math Soc 83(2):309–332, 2011). The generalization we get is to arbitrary language, with no restrictions on the topology [we do not demand type-definabilty of the open set in the definition of essential 1-basedness from Shami (J Lond Math Soc 83(2):309–332, 2011)]. We conclude that every (possibly uncountable) hypersimple unidimensional theory that is not s-essentially 1-based by means of the forking topology is supersimple. We also obtain a strong version of the above dichotomy in the case where the language is countable.