Generalized independent families and dense sets of Box-Product spaces

[EN] A generalization of independent families on a set S is introduced, based on which various topologies on S can be defined. In fact, the set S with any such topology is homeomorphic to a dense subset of the corresponding box product space (Theorem 2.2). From these results, a general version of the Hewitt-Marczewski-Pondiczery theorem for box product spaces can be established. For any uncountable regular cardinal θ, the existence of maximal generalized independent families with some simple conditions, and hence the existence of irresolvable dense subsets of θ-box product spaces of discrete spaces of small sizes, implies the consistency of the existence of measurable cardinal (Theorem 4.5).