On some recent selective properties involving networks

TL;DR

This paper explores selective properties of certain topological spaces involving networks, providing new examples, establishing limitations, and analyzing their relationships with other properties using advanced set-theoretic methods.

Contribution

It introduces consistent examples of R-,H-, and M-nw-selective spaces, and investigates their properties and interactions with set-theoretic cardinal invariants.

Findings

Constructed uncountable examples of selective spaces.

Proved limitations on constructing non-trivial examples.

Showed product of two H-nw-selective spaces may not be M-nw-selective.

Abstract

In this paper we investigate R-,H-, and M-{\it nw}-selective properties introduced in \cite{BG}. In particular, we provide consistent uncountable examples of such spaces and we define \textit{trivial} R-,H-, and M-{\it nw}-selective spaces the ones with countable net weight having, additionally, the cardinality and the weight strictly less then $cov({\cal M})$, $\frak b$, and $\frak d$, respectively. Since we establish that spaces having cardinalities more than $cov({\cal M})$, $\frak b$, and $\frak d$, fail to have the R-,H-, and M-{\it nw}-selective properties, respectively, non-trivial examples should eventually have weight greater than or equal to these small cardinals. Using forcing methods, we construct consistent countable non-trivial examples of R-{\it nw}-selective and H-{\it nw}-selective spaces and we establish some limitations to constructions of non-trivial examples. Moreover, we consistently prove the existence of two H-{\it nw}-selective spaces whose product fails to be M-{\it nw}-selective. Finally, we study some relations between {\it nw}-selective properties and a strong version of the HFD property.