Topologies and ranks for families of theories in various languages

Topological properties and characteristics of families of theories reflect possibilities of separation of theories and a complexity both for theories and their neighbourhoods. Previously, topologies were studied for families of complete theories, in general case and for a series of natural classes, and for various families of incomplete theories in a fixed language. The ranks were defined and described for complete theories in a given language, for a hierarchy of theories, for families of incomplete theories, for formulae and for a series of natural families of theories, including families of ordered theories, families of theories of permutations and families of theories of abelian groups.
In this paper, we study properties and characteristics for topologies and ranks for families of theories in various languages. It is based on special relations connecting formulae in a given language. These relations are used to define and describe kinds of separations with respect to $T_0$-topologies, $T_1$-topologies and Hausdorff topologies. Besides special relations are used to define and study ranks for families of theories in various languages. Possibilities of values for the rank are described, and these possibilities are characterized in topological terms.