Recent zbMATH articles in MSC 54D10https://zbmath.org/atom/cc/54D102024-03-13T18:33:02.981707ZWerkzeugLarge strongly anti-Urysohn spaces existhttps://zbmath.org/1528.540022024-03-13T18:33:02.981707Z"Juhász, István"https://zbmath.org/authors/?q=ai:juhasz.istvan"Shelah, Saharon"https://zbmath.org/authors/?q=ai:shelah.saharon"Soukup, Lajos"https://zbmath.org/authors/?q=ai:soukup.lajos"Szentmiklóssy, Zoltán"https://zbmath.org/authors/?q=ai:szentmiklossy.zoltanRecall that a topological space \(X\) is called Urysohn if for any \(x,y\in X\) with \(x\neq y\), there are open sets \(U,V\subseteq X\) such that \(x\in U\), \(y\in V\), and \(\overline{U}\cap\overline{V}=\emptyset\). A Hausdorff space \(X\) is called anti-Urysohn if for any nonempty open sets \(U,V\subseteq X\), \(\overline{U}\cap\overline{V}\neq\emptyset\). A Hausdorff space \(X\) is called strongly anti-Urysohn if it has at least two non-isolated points and any two infinite closed subsets intersect. In [\textit{I. Juhász} et al., Topology Appl. 213, 8--23 (2016; Zbl 1352.54004)], it was asked (a) if there is a strongly anti-Urysohn space in ZFC, and (b) if it is consistent that there is a strongly anti-Urysohn space of cardinality \(>\mathfrak c\). In the paper under review, the authors answer these questions affirmatively.
Reviewer: Akira Iwasa (Big Spring)