2021/06/01 by István Juhász, Juhász, István, Saharon Shelah +5
Computer Science · Mathematics · #03E04 #54A25 #54A35 #54D10 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #General Topology (math.GN) #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.2106.00618
openalex publication_date 2021/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
As defined in [1], a Hausdorff space is strongly anti-Urysohn (in short: SAU) if it has at least two non-isolated points and any two infinite closed subsets of it intersect. Our main result answers the two main questions of [1] by providing a ZFC construction of a locally countable SAU space of cardinality 2^\mathfrakc. The construction hinges on the existence of 2^\mathfrakc weak P-points in ω^*, a very deep result of Ken Kunen. It remains open if SAU spaces of cardinality > 2^\mathfrakc could exist, while it was shown in [1] that 2^2^\mathfrakc is an upper bound. Also, we do not know if crowded SAU spaces, i.e. ones without any isolated points, exist in ZFC but we obtained the following consistency results concerning such spaces. (1) It is consistent that \mathfrakc is as large as you wish and there is a locally countable and crowded SAU space of cardinality \mathfrakc+. (2) It is consistent that both \mathfrakc and 2^\mathfrakc are as large as you wish and there is a crowded SAU space of cardinality 2^\mathfrakc. (3) For any uncountable cardinal κ the following statements are equivalent: (i) κ=cof([κ]ω,⊆). (ii) There is a locally countable and crowded SAU space of size κ in the generic extension obtained by adding κ Cohen reals. (iii) There is a locally countable and countably compact T1-space of size κ in some CCC generic extension. [1] I. Juhasz, L. Soukup, and Z. Szentmiklossy, Anti-Urysohn spaces, Top. Appl., 213 (2016), pp. 8--23.