2015/09/04 by István Juhász, Juhász, István, Lajos Soukup +3
Mathematics · #03E04 #54A25 #54A35 #54D10 #FOS: Mathematics #General Topology (math.GN) #Logic (math.LO) #math.GN #math.LO #msc:03E04 #msc:54A25 #msc:54A35 #msc:54D10
paper · pdf · doi:10.48550/arxiv.1509.01420
arxiv created 2015/09/04 · arxiv updated 2015/09/07
All spaces are assumed to be infinite Hausdorff spaces. We call a space "anti-Urysohn" (AU in short) iff any two non-emty regular closed sets in it intersect. We prove that \bullet for every infinite cardinal κ there is a space of size κ in which fewer than cf(κ) many non-empty regular closed sets always intersect; \bullet there is a locally countable AU space of size κ iff ω≤ κ≤ 2\mathfrak c. A space with at least two non-isolated points is called "strongly anti-Urysohn" (SAU in short) iff any two infinite closed sets in it intersect. We prove that \bullet if X is any SAU space then \mathfrak s≤ |X|≤ 2^2\mathfrak c; \bullet if \mathfrak r=\mathfrak c then there is a separable, crowded, locally countable, SAU space of cardinality \mathfrak c; \item if λ> ω Cohen reals are added to any ground model then in the extension there are SAU spaces of size κ for all κ∈ [ω1,λ]; \bullet if GCH holds and κ≤λ are uncountable regular cardinals then in some CCC generic extension we have \mathfrak s=κ, \mathfrak c=λ, and for every cardinal μ∈ [\mathfrak s, \mathfrak c] there is an SAU space of cardinality μ. The questions if SAU spaces exist in ZFC or if SAU spaces of cardinality > \mathfrak c can exist remain open.