To the Question on Some Generalizations of Properties of the Linkedness of Families of Sets and the Supercompactness of Topological Spaces

Abstract

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.