#studyprep

02-20

02/10/19-

Today’s to-do list and a TBR list, all for a snowy day. (Gudetama’s such a mood)

The former is a special case of the latter. also note that this is just DST prep for a potential meetup with a mathematician in Finland; they seem pretty strong in DST; obviously not 100% done.

Theorem 1. (Strong Separation Theorem) Suppose κ is an infinite cardinal and A, B are disjoint κ-Suslin subsets of some perfect product space χ. Then there exists a (κ+1)-Borel set C which separates A from B.

Proof (Sketch-ish). This proof simply relies on separating sets combinatorically; we use trees to accomplish this. Basically we use a tree to exhibit a (κ+1)-Borel set C that separates A from B. Suppose that A = ⋃_{i ∈ I} A_i, and B = ⋃_{j ∈ J} B_j. Note that the index sets I and J are arbitrary, and that for each i ∈ I and j ∈ J, there is a set C_{i,j} which separates A_i from B_j. We will then know that C = ⋃_{i ∈ I} ⋂_{j ∈ J} C_{i,j} separates A from B. If it is assumed that A and B are κ-Suslin sets of irrationals, then, at least under closure under Borel substitution and the perfect set theorem/property, we can study κ-Suslin sets in terms of trees. Borel substitution helps in that, given an arbitrary pointset P = 𝒜^κ_u P_u, with each P_u closed, and 𝒜 representing continuous Borel substitution, then f^-1[P] = 𝒜^κ_u. Since each f^-1[P_u] is Borel here, this is κ-Suslin. The perfect set theorem/property helps in that it (especially the theorem's results) holds for non-empty perfect subsets in particular, thus making it "nicer" to work with trees for this.

Not to get too detailed about the tree here, but here is the gist of it; the tree J on ω ⨉ κ ⨉ κ is defined as:

((t_0, ξ_0, η_0),...,(t_{n-1}, ξ_{n-1}, η_{n-1}) ∈ J iff (t_0, ξ_0,...,t_{n-1}, ξ_{n-1}) ∈ T and (t_0, η_0,..., t_{n-1}, η_{n+1})∈S.

And then one can define on the tree a function f: u -> C_u (in which u is a sequence from tree J) such that for each sequence u in ω ⨉ κ ⨉ κ:

(a) C_u is (κ+1)-Borel, and

(b) C_u separates A_τ(u) from B_σ(u),