Topologies, ranks, and closures for families of theories. I

We describe topological properties, ranks, closures, and their dynamics for families of theories. Types of topologies for families of theories are characterized. A relationship is established between ranks and topologies for families of theories. Boolean combinations of $s$-definable families of theories are treated, ranks and degrees with respect to these families are found, and values of the characteristics in question are described. We study closures of families of theories with respect to $s$-definable subfamilies and their Boolean combinations, properties of closure operators, and also a condition for the existence of a least generating set. Rank values for families of theories are specified in terms of algebras of definable subfamilies.