Characters of countably tight spaces and inaccessible cardinals

In this paper, we study some connections between characters of countably tight spaces of size ω 1 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) indestructibly countably tight space of size ω 1 has character ⩽ ω 1 . > ω 2 and there is no countably tight space of size ω 1 and character ω 2 . he converse, we show that, if ω 2 is not inaccessible in the constructible universe L, then there is an indestructibly countably tight space of size ω 1 and character ω 2 .