Characters of countably tight spaces and inaccessible cardinals

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)2^ω1>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.