In this paper, we consider natural generalizations of properties of the linkedness of families (of sets) and the supercompactness of topological spaces. In the first case, we analyze the “multiple” linkedness, assuming the nonemptiness of the intersection of sets from subfamilies, whose cardinality does not exceed some given positive integer \(\mathbf{n}\). In the second case, we study the question of the existence of an (open) prebase such that any its covering has a subcovering, whose cardinality does not exceed \(\mathbf{n}\). We consider maximal \(\mathbf{n}\)-linked (in the mentioned sense) subfamilies of a \(\pi\)-system with “zero” and “unit” (a \(\pi\)-system is a nonempty family closed with respect to finite intersections); these subfamilies are said to be maximal \(\mathbf{n}\)-linked systems or (for short) \(\mathbf{n}\)-MLS. We are interested in correlations between \(\mathbf{n}\)-MLS and ultrafilters (u/f) of a \(\pi\)-system, including the “dynamics” in dependence of \(\mathbf{n}\). Moreover, we consider bitopological spaces (BTS), whose elements are \(\mathbf{n}\)-MLS and u/f; in both cases, for constructing a BTS (a nonempty set with a pair of comparable topologies) we use topologies of Wallman and Stone types. The Wallman-type topology on the set of \(\mathbf{n}\)-MLS realizes an \(\mathbf{n}\)-supercompact (in the sense mentioned above) T₁-space which represents an abstract analog of a superextension of a T₁-space. We prove that the BTS of u/f of the initial \(\pi\)-system is a subspace of the BTS whose points are \(\mathbf{n}\)-MLS; i. e., the corresponding “Wallman” and “Stone” topologies on the set of u/f are induced by the corresponding topologies on the set of \(\mathbf{n}\)-MLS.

在本文中，我们考虑了家庭（集合）的链接性质和拓扑空间的超紧缩性质的自然概括。在第一种情况下，我们假设子集的基数不超过给定正整数\（\ mathbf {n} \）的子族的交集的非空性，我们分析了“多重”链接。在第二种情况下，我们研究了一个（开放的）预基的存在性问题，以便它的任何覆盖都具有子覆盖，其基数不超过\（\ mathbf {n} \）。我们考虑具有（零）和“单位”（\（\ pi \）的\（\ pi \）系统的最大\（\ mathbf {n} \）链接的子族（在上述意义上）-system是相对于有限交叉点而言是封闭的非空族）；这些子族被称为最大\（\ mathbf {n} \）链接系统或（简称）\（\ mathbf {n} \）- MLS。我们对\（\ mathbf {n} \）- MLS与\（\ pi \）系统的超滤器（u / f）之间的相关性感兴趣，包括依赖于\（\ mathbf {n}的“动力学” \）。此外，我们考虑位空间（BTS），其元素为\（\ mathbf {n} \）- MLS和u / f；在这两种情况下，为了构造BTS（具有一对可比较拓扑的非空集），我们都使用Wallman和Stone类型的拓扑。\（\ mathbf {n} \）- MLS集合上的Wallman型拓扑实现了\（\ mathbf {N} \） -supercompact（在这个意义上上面提到的）Ť _1个-space其表示的superextension的抽象模拟Ť ₁ -space。我们证明初始\（\ pi \）- system的u / f的BTS是其点为\（\ mathbf {n} \）- MLS的BTS的子空间。一世。例如，u / f集合上的相应“ Wallman”和“ Stone”拓扑是由\（\ mathbf {n} \）- MLS上的相应拓扑诱导的。