2024/07/18 by Nathan Carlson, Carlson, Nathan
Engineering · Mathematics · Physics and Astronomy · #54A25 #54D10 #Advanced Mathematical Theories and Applications #FOS: Mathematics #General Topology (math.GN) #Graph theory and applications #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2407.13508
openalex publication_date 2024/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathfrakc, 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)≤min\aL(X),e(X)\ and use the Erdős-Rado theorem to show that |X|≤ 2Ue(X)Δ(X) for any Urysohn space X.