To question on some generalizations of properties of cohesion of families of sets and supercompactness of topological spaces

A. G. Chentsov

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2020), pp. 65-80

Résumé

Natural generalizations of the cohesion of families (of sets) and supercompactness of topological spaces are considered. In the first case, “multiple” adhesion is analyzed, when the nonemptyness of intersection of sets from subfamilies with cardinality not more than given natural number $\mathbf{n}$ is postulated, and, in the second case, it is postulated the existence of the (open) subbasis for which any covering has a subcovering with cardinality not more than $\mathbf{n}$. The maximal $\mathbf{n}$-linked (in the above-mentioned sense) subfamilies of a $\pi$-system with “zero” and “unit” are investigated; these subfamilies are called maximal $\mathbf{n}$-linked systems or (briefly) $\mathbf{n}$-MLS. Relations between $\mathbf{n}$-MLS and ultrafilters (u/f) of a $\pi$-system are investigated including “dynamics” under variation of $\mathbf{n}$. Moreover, bitopological spaces (BTS) with elements in the form of $\mathbf{n}$-MLS and u/f are investigated; as topologies used under construction of BTS (this is a nonempty set with pair of comparable topologies), in both cases, topologies of Wallman type and Stone type apply. In addition, topology of Wallman type on the set of $\mathbf{n}$-MLS realizes a $\mathbf{n}$-supercompact (in the above-mentioned sense) $T_1$-space; this space is analog of superextension of $T_1$-space. It is demonstrated that BTS of u/f of the initial $\pi$-system is a subspace of BTS with points in the form of $\mathbf{n}$-MLS: the corresponding “Wallman” and “Stone” topologies on the u/f set are induced by the corresponding topologies on the set of $\mathbf{n}$-MLS.

Détail

Keywords: bitopological space, linked system, ultrafilters.