625 Thackeray Hall
Abstract or Additional Information
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(X), where e(X)≤sn(X)≤L(X), is used to establish new cardinal inequalities involving diagonal degrees. We show |X|≤sn(X)Δ(X) for a T1 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 Gδ-diagonal has cardinality at most continuum, as well as three results of Gotchev. Further results lead to logical improvements of theorems of Basile, Bella, and Ridderbos and a partial solution to a question of the same authors. Finally, we define the Urysohn extent Ue(X) with the property Ue(X)≤min 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.