The Lifting Problem with the Full Ideal
نویسنده
چکیده
We show that there are a cardinal μ, a σ-ideal I ⊆ P(μ) and a σ-subalgebra B of subsets of μ extending I such that B/I satisfies the c.c.c. but the quotient algebra B/I has no lifting. 0. Introduction. In the present paper we prove the following theorem. Theorem 0.1. For some μ (in fact, μ = (2א0)++ suffices) there is a σ-ideal I on P(μ) and a σ-subalgebra B of P(μ) extending I such that B/I satisfies the c.c.c. but B/I has no lifting. This result answers a question of David Fremlin (see chapter on measure algebras in Fremlin [2]). Moreover, it solves the problem of topologizing a Category Base (see Detlefsen Szymański [3], Morgan [6], Shilling [11] and Szymański [12]). Note that it is well known (Mokobodzki’s theorem; see Fremlin [2]) that under CH, if |B/I| ≤ (2א0)+ then this is impossible; i.e. the quotient algebra B/I has a lifting. Toward the end we deal with having better μ. I thank Andrzej Szymański for asking me the question and Max Burke and Mariusz Rabus for corrections. Notation: Our notation is rather standard. All cardinals are assumed to be infinite and usually they are denoted by λ, κ, μ. In Boolean algebras we use ∩ (and ⋂
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تاریخ انتشار 1997