Characters of countably tight spaces and inaccessible cardinals

• Published in: Topology and its applications Vol. 161; pp. 95 - 106
• Main Author: Usuba, Toshimichi
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.01.2014
• Subjects: Countable tight space; Countable tightness indestructibility; Inaccessible cardinal; Kurepa tree; Topological game; Countable tight space; Countable tightness indestructibility; Inaccessible cardinal; Kurepa tree; Topological game; primary
• ISSN: 0166-8641; 1879-3207
• DOI: 10.1016/j.topol.2013.09.011

In this paper, we study some connections between characters of countably tight spaces of size ω1 and inaccessible cardinals. A countable tight space is indestructible if every σ-closed forcing notion preserves countable tightness of the space. We show that, assuming the existence of an inaccessible cardinal, the following statements are consistent:(1)Every indestructibly countably tight space of size ω1 has character ⩽ω1.(2)2ω1>ω2 and there is no countably tight space of size ω1 and character ω2. For the converse, we show that, if ω2 is not inaccessible in the constructible universe L, then there is an indestructibly countably tight space of size ω1 and character ω2.