On partitions of the real line into compact sets

Extract

The problem mentioned in the title has already been investigated by J. Baumgartner, J. Stern, A. Miller and many others (see [2] and [5]). We prove here some generalizations of theorems of Miller and Stern from [2] and [5]. We use standard set-theoretical notation. Let

One can check that in the above definition we can replace “compact subset of ^ωω” by “closed nowhere dense subset of ^ω2” or “F_σ and meager subset of ^ω2” (as any F_σ subset of ^ω2 can be presented as a disjoint countable union of compact sets).

For functions f, g ϵ ^ωω we define f ≼ g if for all but finitely many n ϵ ω we have f(n) ≤ g(n). Let denote the least cardinality of a family A ⊆ ^ωω such that for any f ϵ ^ωω there is g ϵ A for which f ≼ g. It is easy to see that ≤ κ_ω ≤ κ₁. If f ϵ ^ωω then let ≼(f) = {h ϵ^ωω: h ≼ f}.

We find an axiom which implies = ω₁ → κ₁ = ω₁, and which can be preserved by any ccc notion of forcing of “small cardinality”. We construct also in a generic model many partitions of ^ωω into compact sets preserved not only by any random real extension, but also by Sacks' notion of forcing. This shows that from some point of view Miller's modification of Sacks' forcing (from [2]) is the “minimal” one able to destroy a partition of ^ωω into compact sets.