On diagonal degrees and star networks

Given an open cover $U$ of a topological space X, we introduce the notion of a star network for $U$. The associated cardinal function $sn\left(X\right)$, where $e\left(X\right)\le sn\left(X\right)\le L\left(X\right)$, is used to establish new cardinal inequalities involving diagonal degrees. We show $\left|X\right|\le sn{\left(X\right)}^{\Delta \left(X\right)}$ 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 $sn\left(X\right)$. One result has as corollaries Buzyakova's theorem that a ccc space with a regular $G_{\delta }$-diagonal has cardinality at most $c$, 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 $Ue\left(X\right)$ with the property $Ue\left(X\right)\le \min \left\{aL\right(X),e(X\left)\right\}$ and use the Erdős-Rado theorem to show that $\left|X\right|\le 2^{Ue\left(X\right)\bar{\Delta }\left(X\right)}$ for any Urysohn space X.