Fallen cardinals Original Research Article

We prove that for every singular cardinal μ of cofinality ω, the complete Boolean algebra View the MathML source contains a complete subalgebra which is isomorphic to the collapse algebra View the MathML source. Consequently, adding a generic filter to the quotient algebra View the MathML source collapses View the MathML source to View the MathML source. Another corollary is that the Baire number of the space U(μ) of all uniform ultrafilters over μ is equal to View the MathML source. The corollaries affirm two conjectures of Balcar and Simon. The proof uses pcf theory.