How do we know that the quotient of the Boolean algebra associated with Cohen forcing by a generic filter is either atomic or isomorphic to the Cohen forcing? I know that Cohen forcing is the unique separable atomless complete Boolean algebra, but then why is the quotient either atomic or atomless?
Thanks
A quotient (homomorphic image) of Cohen algebra may not be atomless or purely atomic. Although you can find a maximal antichain A in a quotient B such that each a in A is an atom or B restricted to a is atomless - this is true of every separable algebra and quotients are clearly separable.