On diagonal degrees and star networks

TL;DR

This paper introduces a new cardinal function called star network for open covers, establishing inequalities that relate the size of topological spaces to diagonal degrees and other invariants, advancing understanding of space cardinalities.

Contribution

The paper defines the star network cardinal function and uses it to derive new inequalities involving diagonal degrees, partially answering longstanding questions and improving existing theorems.

Findings

Proves $|X| extless= sn(X)^{ riangle(X)}$ for $T_1$ spaces.

Derives bounds on space cardinality using star network and Urysohn extent.

Provides partial solutions and improvements to classical topological theorems.

Abstract

Given an open cover $\mathcal{U}$ of a topological space $X$, we introduce the notion of a star network for $\mathcal{U}$. The associated cardinal function $sn(X)$, where $e(X)\leq sn(X)\leq L(X)$, is used to establish new cardinal inequalities involving diagonal degrees. We show $|X|\leq sn(X)^{\Delta(X)}$ for a $T_1$ space $X$, giving a partial answer to a long-standing question of Angelo Bella. Many further results are given using variations of $sn(X)$. One result has as corollaries Buzyakova's theorem that a ccc space with a regular $G_\delta$-diagonal has cardinality at most $\mathfrak{c}$, as well as three results of Gotchev. Further results lead to logical improvements of theorems of Basile, Bella, and Ridderbos, a partial solution to a question of the same authors, and a theorem of Gotchev, Tkachenko, and Tkachuk. Finally, we define the Urysohn extent $Ue(X)$ with the property $Ue(X)\leq\min\{aL(X),e(X)\}$ and use the Erd\H{o}s-Rado theorem to show that $|X|\leq 2^{Ue(X)\overline{\Delta}(X)}$ for any Urysohn space $X$.