LAUSR

To the best of our knowledge, all the existing variational principles were established in the framework of a complete metric space or a Banach space. In this paper, we consider variational principles in topological spaces. We adopt gauge-type functions on a topological space X and introduce the notion of the Cantor compatibility for X and gauge-type functions on X. In terms of the Cantor-compatibility, we provide strong variational principles on a topological space, which further extend both the Ekeland variational principle and the Borwein-Preiss smooth variational principle.