Maddalena Bonanzinga - ATTI 2026 | Algebra, Topology and Their Interactions

Some selection principles defined by networks.

In [Sch1] a new systematic description of covering properties in terms of selection principles has been introduced. Later, this approach has been extended to non-covering properties. Given two collections $\cal A$ and $\cal B$ of some particular subfamilies or subsets of a space $X$ , Scheepers introduce the following notation:

[ $S_1({\cal A},{\cal B})$ ]: For every sequence $({\cal U}_n: n\in{\Bbb N})$ of elements of $\cal A$ there exists $U_n\in {\cal U}_n$ , $n\in{\Bbb N}$ , such that $\{U_n:n\in{\Bbb N}\}$ belongs to $\cal B$ .
[ $S_{fin}({\cal A},{\cal B})$ ]: For every sequence $({\cal U}_n: n\in{\Bbb N})$ of elements of $\cal A$ there exists a finite subset ${\cal F}_n\subseteq {\cal U}_n$ , $n\in{\Bbb N}$ , such that $\bigcup_{n\in{\Bbb N}} {\cal F}_n$ belongs to $\cal B$ .
[ $U_{fin}({\cal A},{\cal B})$ ]: For every sequence $({\cal U}_n: n\in{\Bbb N})$ of elements of $\cal A$ there exists a finite subset ${\cal F}_n\subseteq {\cal U}_n$ , $n\in{\Bbb N}$ , such that $\{\bigcup{\cal F}_n : n\in{\Bbb N}\}$ belongs to $\cal B$ .

Recall that a $\gamma$ -cover of a space $X$ a particular cover such that each point of $X$ belongs to all but finitely many members of the cover. If ${\cal O}$ is the family of all open covers and $\Gamma$ is the family of all $\gamma$ -covers of a space $X$ , then the selection pricinples $S_1({\cal O},{\cal O})$ , $S_{fin}({\cal O},{\cal O})$ $U_{fin}({\cal O},\Gamma)$ , denote Rothberger, Menger and Hurewicz property, respectively.

A space $X$ is M-separable (see [BBM]) if it satisfies the selection principle $S_{fin}({\cal D},{\cal D})$ , where ${\cal D}$ is the collection of all dense subsets of $X$ . Every space having a countable base is M-separable but not every space with countable network weight is M-separable. We present a new Menger-type property defined by networks, called M-nw-selective property, such that every M-nw-selective space has countable network weight and is M-separable. By analogy, H- and R- nw-selective spaces for Hurewicz and Rothberger type properties are considered.
In [BG, BGZ] several properties of the new classes of spaces are studied and some questions are posed.

Also, in [ABG] the new topological games called R-nw-selective game and the M-nw-selective game that naturally arise from the corresponding selection principles involving networks are introduced and investigated.

[Sch1] M. Scheepers, Combinatorics of open covers I: Ramsey Theory, Topology and its Applications, 69 (1996) 31-62.

[BBM] A. Bella, M. Bonanzinga, M.V. Matveev, Variations of selective separability, Topology and its Applications, 156 (2009) 1241-1252.

[BG] M. Bonanzinga, D. Giacopello, A generalization of M-separability by networks, Atti della Accademia Peloritana dei Pericolanti, 101, 2, A11 (2023).

[ABG] L. F. Aurichi, M. Bonanzinga. D. Giacopello, On some topological games involving networks, Topology and its Applications, accepted.

[BGZ] M. Bonanzinga, D. Giacopello, L. Zdomskyy, On M-nw-selection, preprint.