2023/07/20 by István Juhász, Juhász, István, Lajos Soukup +3
Computer Science · Mathematics · #03E35 #54A25 #54A35 #54D99 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #General Topology (math.GN) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2307.11014
openalex publication_date 2023/07/20 · openalex created_date 2023/07/22 · openalex updated_date 2026/07/28
Hart and Kunen, and independently in the recent preprint arXiv:2304.13113, Ríos-Herrejón defined and studied the class C(ω1) of topological spaces X having the property that for every neighborhood assignment \U(y) : y ∈ Y\ with Y ∈ [X]ω1 there is Z ∈ [Y]ω1 such that Z ⊂ \bigcap \U(z) : z ∈ Z\. It is obvious that spaces of countable net weight, i.e. having a countable network, belong to this class. In this paper we present several independence results concerning the relationships of these and several other classes that are sandwiched between them. These clarify some of the main problems that were raised in the above preprint. In particular, we prove that the continuum hypothesis, in fact a weaker combinatorial principle called super stick, implies that every regular space in C(ω1) has countable net weight, answering a question that was raised by Hart and Kunen.