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.

我们针对理论族描述拓扑属性，等级，闭包及其动态。表征了理论族的拓扑类型。在理论族的等级和拓扑之间建立了关系。定义了s可定义理论族的布尔组合，找到了相对于这些族的等级和度，并描述了所讨论特征的值。我们研究关于s可定义的子族及其布尔组合的理论族的闭包，闭包算子的性质，以及存在最小生成集的条件。理论族的等级值是根据可定义的亚族的代数来指定的。