On Diagonal Degrees and Star Networks

Thursday, September 5, 2024 - 11:30

625 Thackeray Hall

HideSpeaker Information
Nathan Carlson
California Lutheran University

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.