Baire number of the spaces of uniform ultrafilters

The Baire number is defined for a topological space without isolated points as the minimal size of the family of nowhere dense sets covering the space in question. We prove that in the case ofU(κ), the space of uniform ultrafilters over uncountable κ, the Baire number equals eitherω ₁ orω ₂, depending on the cofinality of κ. The results are connected to the collapsing of cardinals when using the quotient algebraP(κ) modκ]^<κ as the notion of forcing. The main portion of the present research, was done at the Center for Theoretical Study at Charles University and the Academy of Sciences.