Representing sets for σ-ideals of compact sets
Maya Saran
Ashoka University
June 24, 2014
Outline of this talk
1. Definitions and background
2. Solecki’s representation theorem for ideals of compact sets
3. A refinement of the representation theorem
The setting
E a compact metric space.
K(E ) = {F ⊆ E : F compact}
Hausdorff metric dH on K(E ) satisfies, for nonempty F , K ∈ K(E ),
dH (F , K ) < ⇐⇒ F ⊆ B(K , ) and K ⊆ B(F , )
S
where B(K , ) = x∈K B(x, ).
This makes K(E ) into a compact metric space.
Definitions: ideals and σ-ideals of compact sets
A set I ⊆ K(E ) is an ideal of compact sets if it is:
1. downward closed: if F1 ⊆ F2 and F2 ∈ I then F1 ∈ I
S
2. closed under finite union: if F1 , . . . , Fk ∈ I then k1 Fi ∈ I .
An ideal I is a σ-ideal of compact sets if it is also closed under
countable unions whenever the union itself is compact.
Examples
Ideals of compact sets arise out of various notions of smallness.
Examples include ideals of the following compact sets:
I
meager sets
I
null sets for a finite Borel measure
I
sets of dimension ≤ n for fixed n ∈ N
I
zero sets with respect to a Hausdorff measure
I
Z-sets for E = [0, 1]ω
Complexity of an ideal
The condition of being an ideal or σ-ideal is strongly related to the
complexity of I ⊆ K(E ).
I
Kechris–Louveau, Dougherty: if I is a Gδ ideal, it must be a
σ-ideal
I
Kechris–Louveau–Woodin: if a σ-ideal I is either co-analytic
or analytic, it must be either complete co-analytic or simply
Gδ .
Property (∗)
We consider Gδ ideals of compact sets that satisfy the following
natural condition, formulated by Solecki:
A set I ⊆ K (E ) has property (∗) if, for any
Ssequence of sets
Kn ∈ I , there exists a Gδ set G such that n Kn ⊆ G and
K(G ) ⊆ I .
Easy to see: if I ⊆ K (E ) has property (∗), it must be a σ-ideal.
Comments on property (∗)
If I has (∗) and is analytic or co-analytic, it must be a Gδ σ-ideal.
Longstanding question: Does every Gδ ideal have property (∗)?
Recently settled by Matrai – no.
Property (∗) does however hold in all natural examples of Gδ ideals
(including all examples mentioned.)
A representation theorem
For A ⊆ E , define
A∗ = {K ∈ K(E ) : K ∩ A 6= ∅}
Theorem (Solecki). Suppose I is co-analytic and non-empty.
Then I has property (∗) iff there exists a closed set F ⊆ K(E )
such that, for any K ∈ K(E ),
K ∈ I ⇐⇒ K ∗ ∩ F is meager in F.
Example: Lebesgue measure on the plane
For E the unit square in the plane and λ Lebesgue measure on E ,
let
I = {K ∈ K(E ) : λ(K ) = 0}
Then the set
F = {K ∈ K(E ) : λ(K ) ≥ 1/2}
works to characterize membership in the ideal.
Comments on the representative
F is not unique. Are there properties for F that make it a
canonical representative (perhaps upto some notion of
equivalence)?
We call F upward closed if
∀A, B ∈ K(E ), B ⊇ A ∈ F ⇒ B ∈ F
If F is closed upwards, the map K 7→ K ∗ ∩ F, a fundamental
function in this context, is continuous.
Result on finding an upward closed representative
Theorem (S.) Suppose I ⊆ K(E ) is a nonempty co-analytic set
with property (∗), and suppose that all sets in I have empty
interior. Then there exists a closed set F ⊆ K(E ) such that F is
upward closed and, for any K ∈ K(E ),
K ∈ I ⇐⇒ K ∗ ∩ F is meager in F.
Result in terms of closed sets of K(E )
Theorem (S.) For a closed set F ⊆ K(E ), the following are
equivalent:
(1) ∀K ∈ K(E ), K has nonempty interior ⇒ K ∗ nonmeager in F.
(2) ∃F 0 ⊆ K(E ), closed and closed upwards, such that
∀K ∈ K(E ), K ∗ nonmeager in F 0 ⇐⇒ K ∗ nonmeager in F.
Idea of proof
Let F be a closed subset of K(E ) such that
I = {K ∈ K(E ) : K ∗ is meager in F}
Let {Vn } be a basis for the relative topology on F.
Let {Un } be a sequence of open disjoint sets in E ; associate Vn
with Un . Define F 0 ⊆ K(E ) such that sets in F 0 must contain, for
some n, a set in Vn and a point in Un . By excluding some Un ’s
from an open subset of F, we exclude the corresponding Vn ’s.
Another direction
Given a set F ⊆ K(E ), we may consider Gδ sets G ⊆ E for which
G ∗ is meager in F.
Meagerness of G ∗ in F typically does not correspond to smallness
of the set G .
Question:
For which Gδ sets G ⊆ E will we have G ∗ meager in F?
References
É. Matheron and M. Zelený, Descriptive set theory of families of
small sets, Bull. Symbolic Logic 13 (2007), no. 4, 482–537
S. Solecki, Gδ Ideals of Compact Sets, J. Eur. Math. Soc. 13
(2011), 853–882
M. Saran, A Note on Gδ Ideals of Compact Sets, Comment. Math.
Univ. Carolinae, 50, 4 (2009) 569-573