Forceless, ineffective, powerless proofs of
descriptive set-theoretic dichotomy theorems
Benjamin D. Miller
Boise extravaganza in set theory
March 27th , 2009
Part I
A selective history
I. A selective history
The late 19th century
I. A selective history
The late 19th century
The birth of set theory
The roots of set theory first sprout with Cantor’s realization that
there is no injection of R into N.
I. A selective history
The late 19th century
The birth of set theory
The roots of set theory first sprout with Cantor’s realization that
there is no injection of R into N.
Fruits of the continuum hypothesis
While trying to establish his continuum hypothesis, Cantor proves
the first descriptive set-theoretic dichotomy theorem.
I. A selective history
The late 19th century
The birth of set theory
The roots of set theory first sprout with Cantor’s realization that
there is no injection of R into N.
Fruits of the continuum hypothesis
While trying to establish his continuum hypothesis, Cantor proves
the first descriptive set-theoretic dichotomy theorem.
Theorem (Cantor)
Suppose that X is a closed subset of R. Then exactly one of the
following holds:
The set X is countable.
There is a continuous injection of 2ω into X .
I. A selective history
The early 20th century
I. A selective history
The early 20th century
The birth of descriptive set theory
The study of definable subsets of R begins with the investigations
of Baire, Borel, and Lebesgue into Borel sets.
I. A selective history
The early 20th century
The birth of descriptive set theory
The study of definable subsets of R begins with the investigations
of Baire, Borel, and Lebesgue into Borel sets.
Definition (Souslin)
A Hausdorff space X is analytic if it is the continuous image of a
closed subset of ω ω .
I. A selective history
The early 20th century
I. A selective history
The early 20th century
Theorem (Lusin, Souslin)
Disjoint analytic sets can be separated by a Borel set. In particular,
every bi-analytic subset of a Hausdorff space is Borel, and a subset
of an analytic Hausdorff space is Borel if and only if it is bi-analytic.
I. A selective history
The early 20th century
Theorem (Lusin, Souslin)
Disjoint analytic sets can be separated by a Borel set. In particular,
every bi-analytic subset of a Hausdorff space is Borel, and a subset
of an analytic Hausdorff space is Borel if and only if it is bi-analytic.
Theorem (Souslin)
Suppose that X is an analytic Hausdorff space. Then exactly one
of the following holds:
The set X is countable.
There is a continuous injection of 2ω into X .
I. A selective history
The early 20th century
I. A selective history
The early 20th century
Theorem (Lusin-Sierpiński)
Every analytic subset of a Polish space has the Baire property.
I. A selective history
The early 20th century
Theorem (Lusin-Sierpiński)
Every analytic subset of a Polish space has the Baire property.
Theorem (Lusin-Novikov)
Suppose that X and Y are Hausdorff spaces and R ⊆ X × Y is analytic. Then exactly one of the following holds:
There are partial functions fn ··· XS → Y , whose graphs are Borel subsets of R, such that R = n∈ω graph(fn ).
For some x ∈ X , there is a continuous injection of 2ω into Rx .
I. A selective history
The 1970s
I. A selective history
The 1970s
Definition
Suppose that R ⊆ X × X and S ⊆ Y × Y . A reduction of R to S
is a map π : X → Y such that R = (π × π)−1 (S). An embedding
of R into S is an injective reduction of R to S.
I. A selective history
The 1970s
Definition
Suppose that R ⊆ X × X and S ⊆ Y × Y . A reduction of R to S
is a map π : X → Y such that R = (π × π)−1 (S). An embedding
of R into S is an injective reduction of R to S.
Theorem (Silver)
Suppose that X is a Hausdorff space and E is a co-analytic equivalence relation on X . Then exactly one of the following holds:
The set X /E is countable.
There is a continuous embedding of ∆(2ω ) into E .
I. A selective history
The 1970s
I. A selective history
The 1970s
Remark
Silver’s proof was quite difficult and used a large fragment of ZF.
Harrington later gave a simpler proof using the effective theory.
I. A selective history
The 1970s
Remark
Silver’s proof was quite difficult and used a large fragment of ZF.
Harrington later gave a simpler proof using the effective theory.
Remark
Since then, a variety of perfect-set-style dichotomy theorems have
been established using Harrington’s ideas.
I. A selective history
The 1970s and 1980s
I. A selective history
The 1970s and 1980s
Remark
Examples include basis theorems for the classes of uncountable
quasi-orders, linear quasi-orders, quasi-metric spaces, and vector
spaces which are suitably definable.
I. A selective history
The 1970s and 1980s
Remark
Examples include basis theorems for the classes of uncountable
quasi-orders, linear quasi-orders, quasi-metric spaces, and vector
spaces which are suitably definable.
Remark
Over the last twenty years, a number of more sophisticated dichotomy theorems have been discovered.
I. A selective history
The 1990s
I. A selective history
The 1990s
Definition
Let E0 denote the equivalence relation on 2ω given by
xE0 y ⇐⇒ ∃n ∈ ω (x (n, ∞) = y (n, ∞)).
I. A selective history
The 1990s
Definition
Let E0 denote the equivalence relation on 2ω given by
xE0 y ⇐⇒ ∃n ∈ ω (x (n, ∞) = y (n, ∞)).
Theorem (Harrington-Kechris-Louveau)
Suppose that X is a Hausdorff space and E is a bi-analytic equivalence relation on X . Then exactly one of the following holds:
There is a Borel reduction of E to ∆(2ω ).
There is a continuous embedding of E0 into E .
Part II
Interlude
II. Interlude
II. Interlude
Remark
Although the statements of these results are purely classical, their
known proofs rely heavily on tools from mathematical logic.
II. Interlude
Remark
Although the statements of these results are purely classical, their
known proofs rely heavily on tools from mathematical logic.
Remark
It is therefore natural to ask whether they have classical proofs.
II. Interlude
Remark
Although the statements of these results are purely classical, their
known proofs rely heavily on tools from mathematical logic.
Remark
It is therefore natural to ask whether they have classical proofs.
Remark
It is reasonable to hope that such proofs could ease the generalization of these theorems to broader classes of definable sets, or even
uncover paths to new theorems.
Part III
A new approach
III. A new approach
III. A new approach
Remark
Another dichotomy theorem of a somewhat different type was discovered in the early 1990s.
III. A new approach
Remark
Another dichotomy theorem of a somewhat different type was discovered in the early 1990s.
Definition
A homomorphism from R to S is a function φ : X → Y such that
(φ × φ)(R) ⊆ S.
III. A new approach
Remark
Another dichotomy theorem of a somewhat different type was discovered in the early 1990s.
Definition
A homomorphism from R to S is a function φ : X → Y such that
(φ × φ)(R) ⊆ S.
Definition
A graph on X is an irreflexive, symmetric set G ⊆ X × X .
III. A new approach
Remark
Another dichotomy theorem of a somewhat different type was discovered in the early 1990s.
Definition
A homomorphism from R to S is a function φ : X → Y such that
(φ × φ)(R) ⊆ S.
Definition
A graph on X is an irreflexive, symmetric set G ⊆ X × X .
Definition
An I -coloring of a graph G is a homomorphism from G to ∆(I )c .
III. A new approach
III. A new approach
Definition (Kechris-Solecki-Todorcevic)
Fix sequences sn ∈ 2n such that ∀s ∈ 2<ω ∃n ∈ ω (s v sn ), and set
[
G0 =
{(sn a i a x, sn a (1 − i)a x) | i ∈ 2 and x ∈ 2ω }.
n∈ω
III. A new approach
Definition (Kechris-Solecki-Todorcevic)
Fix sequences sn ∈ 2n such that ∀s ∈ 2<ω ∃n ∈ ω (s v sn ), and set
[
G0 =
{(sn a i a x, sn a (1 − i)a x) | i ∈ 2 and x ∈ 2ω }.
n∈ω
Theorem (Kechris-Solecki-Todorcevic)
Suppose that X is a Hausdorff space and G is an analytic graph on
X . Then exactly one of the following holds:
There is a Borel ω-coloring of G.
There is a continuous homomorphism from G0 to G.
III. A new approach
III. A new approach
Remark
Their proof utilized very little of the effective theory.
III. A new approach
Remark
Their proof utilized very little of the effective theory.
Important observation
By introducing an appropriate derivative, one can remove their use
of the effective theory and obtain a classical proof of their theorem.
III. A new approach
III. A new approach
Remark
Unfortunately, there are technical reasons which preclude Silver’s
theorem from having a similar classical proof.
III. A new approach
Remark
Unfortunately, there are technical reasons which preclude Silver’s
theorem from having a similar classical proof.
Important observation
Nevertheless, the Kechris-Solecki-Todorcevic theorem itself can be
used to give a classical proof of Silver’s theorem.
III. A new approach
The first classical proof of Silver’s theorem
III. A new approach
The first classical proof of Silver’s theorem
Sketch of the proof
III. A new approach
The first classical proof of Silver’s theorem
Sketch of the proof
Set G = E c .
III. A new approach
The first classical proof of Silver’s theorem
Sketch of the proof
Set G = E c .
Then X /E is countable iff there is an ω-coloring of G.
III. A new approach
The first classical proof of Silver’s theorem
Sketch of the proof
Set G = E c .
Then X /E is countable iff there is an ω-coloring of G.
By the Kechris-Solecki-Todorcevic theorem, we can assume
that there is a continuous homomorphism φ from G0 to G.
III. A new approach
The first classical proof of Silver’s theorem
Sketch of the proof
Set G = E c .
Then X /E is countable iff there is an ω-coloring of G.
By the Kechris-Solecki-Todorcevic theorem, we can assume
that there is a continuous homomorphism φ from G0 to G.
Then F = (φ × φ)−1 (E ) has the Baire property.
III. A new approach
The first classical proof of Silver’s theorem
Sketch of the proof
Set G = E c .
Then X /E is countable iff there is an ω-coloring of G.
By the Kechris-Solecki-Todorcevic theorem, we can assume
that there is a continuous homomorphism φ from G0 to G.
Then F = (φ × φ)−1 (E ) has the Baire property.
Note that G0 C = ∅ for every equivalence class C of F .
III. A new approach
The first classical proof of Silver’s theorem
Sketch of the proof
Set G = E c .
Then X /E is countable iff there is an ω-coloring of G.
By the Kechris-Solecki-Todorcevic theorem, we can assume
that there is a continuous homomorphism φ from G0 to G.
Then F = (φ × φ)−1 (E ) has the Baire property.
Note that G0 C = ∅ for every equivalence class C of F .
It follows that every equivalence class of F is meager.
III. A new approach
The first classical proof of Silver’s theorem
Sketch of the proof
Set G = E c .
Then X /E is countable iff there is an ω-coloring of G.
By the Kechris-Solecki-Todorcevic theorem, we can assume
that there is a continuous homomorphism φ from G0 to G.
Then F = (φ × φ)−1 (E ) has the Baire property.
Note that G0 C = ∅ for every equivalence class C of F .
It follows that every equivalence class of F is meager.
The Kuratowski-Ulam theorem then implies that F is meager.
III. A new approach
The first classical proof of Silver’s theorem
Sketch of the proof
Set G = E c .
Then X /E is countable iff there is an ω-coloring of G.
By the Kechris-Solecki-Todorcevic theorem, we can assume
that there is a continuous homomorphism φ from G0 to G.
Then F = (φ × φ)−1 (E ) has the Baire property.
Note that G0 C = ∅ for every equivalence class C of F .
It follows that every equivalence class of F is meager.
The Kuratowski-Ulam theorem then implies that F is meager.
The Kuratowski-Mycielski theorem therefore ensures the existence of a continuous embedding ψ of ∆(2ω ) into F .
III. A new approach
The first classical proof of Silver’s theorem
Sketch of the proof
Set G = E c .
Then X /E is countable iff there is an ω-coloring of G.
By the Kechris-Solecki-Todorcevic theorem, we can assume
that there is a continuous homomorphism φ from G0 to G.
Then F = (φ × φ)−1 (E ) has the Baire property.
Note that G0 C = ∅ for every equivalence class C of F .
It follows that every equivalence class of F is meager.
The Kuratowski-Ulam theorem then implies that F is meager.
The Kuratowski-Mycielski theorem therefore ensures the existence of a continuous embedding ψ of ∆(2ω ) into F .
The function φ ◦ ψ is as desired.
III. A new approach
A generalization of the G0 dichotomy theorem
III. A new approach
A generalization of the G0 dichotomy theorem
Remark
Similar arguments can be used to prove all of the dichotomy theorems mentioned thus far, with the lone exception of the theorem of
Harrington-Kechris-Louveau.
III. A new approach
A generalization of the G0 dichotomy theorem
Remark
Similar arguments can be used to prove all of the dichotomy theorems mentioned thus far, with the lone exception of the theorem of
Harrington-Kechris-Louveau.
Important observation
Fortunately, the latter result can be proven from a generalization of
the Kechris-Solecki-Todorcevic theorem.
III. A new approach
A generalization of the G0 dichotomy theorem
III. A new approach
A generalization of the G0 dichotomy theorem
Definition
Fix sequences s2n ∈ 22n with the property that
∀s ∈ 2<ω ∃n ∈ ω (s v s2n ).
Set G0even =
S
n∈ω {(s2n
a i a x, s
2n
a (1
− i)a x) | i ∈ 2 and x ∈ 2ω }.
III. A new approach
A generalization of the G0 dichotomy theorem
Definition
Fix sequences s2n ∈ 22n with the property that
∀s ∈ 2<ω ∃n ∈ ω (s v s2n ).
Set G0even =
S
n∈ω {(s2n
a i a x, s
2n
a (1
− i)a x) | i ∈ 2 and x ∈ 2ω }.
Definition
Fix sequences s2n+1 ∈ 22n+1 × 22n+1 with the property that
∀s ∈ 2<ω × 2<ω ∃n ∈ ω∀i ∈ 2 (s(i) v s2n+1 (i)).
S
Set H0odd = n∈ω {(s2n+1 (i)a i a x, s2n+1 (1 − i)a (1 − i)a x) | i ∈
2 and x ∈ 2ω }.
III. A new approach
A generalization of the G0 dichotomy theorem
III. A new approach
A generalization of the G0 dichotomy theorem
Notation
Let TC(R) denote the transitive closure of R.
III. A new approach
A generalization of the G0 dichotomy theorem
Notation
Let TC(R) denote the transitive closure of R.
Theorem
Suppose that X is a Hausdorff space and G and H are analytic
graphs on X . Then exactly one of the following holds:
There are Borel functions φ : X → ω and ψ : X → 2ω such
that φ × ψ is a coloring of G and ψ is a homomorphism from
H to ∆(2ω ).
There is a continuous homomorphism π : 2ω → X from the
pair (G0even , H0odd ) to the pair (G, TC(H)).
III. A new approach
The first classical proof of the E0 dichotomy theorem
III. A new approach
The first classical proof of the E0 dichotomy theorem
Sketch of the proof
III. A new approach
The first classical proof of the E0 dichotomy theorem
Sketch of the proof
Set G = E c and H = E \ ∆(X ).
III. A new approach
The first classical proof of the E0 dichotomy theorem
Sketch of the proof
Set G = E c and H = E \ ∆(X ).
One can construct a Borel reduction of E to ∆(2ω ) from Borel functions φ : X → ω and ψ : X → 2ω for which φ × ψ is a
coloring of G and ψ is a homomorphism from H to ∆(2ω ).
III. A new approach
The first classical proof of the E0 dichotomy theorem
Sketch of the proof
Set G = E c and H = E \ ∆(X ).
One can construct a Borel reduction of E to ∆(2ω ) from Borel functions φ : X → ω and ψ : X → 2ω for which φ × ψ is a
coloring of G and ψ is a homomorphism from H to ∆(2ω ).
By the dichotomy for pairs, we can assume that there is a
continuous homomorphism φ from (G0even , H0odd ) to (E c , E ).
III. A new approach
The first classical proof of the E0 dichotomy theorem
Sketch of the proof
Set G = E c and H = E \ ∆(X ).
One can construct a Borel reduction of E to ∆(2ω ) from Borel functions φ : X → ω and ψ : X → 2ω for which φ × ψ is a
coloring of G and ψ is a homomorphism from H to ∆(2ω ).
By the dichotomy for pairs, we can assume that there is a
continuous homomorphism φ from (G0even , H0odd ) to (E c , E ).
Then F = (φ × φ)−1 (E ) has the Baire property.
III. A new approach
The first classical proof of the E0 dichotomy theorem
Sketch of the proof
Set G = E c and H = E \ ∆(X ).
One can construct a Borel reduction of E to ∆(2ω ) from Borel functions φ : X → ω and ψ : X → 2ω for which φ × ψ is a
coloring of G and ψ is a homomorphism from H to ∆(2ω ).
By the dichotomy for pairs, we can assume that there is a
continuous homomorphism φ from (G0even , H0odd ) to (E c , E ).
Then F = (φ × φ)−1 (E ) has the Baire property.
The fact that G0even ∩ F = ∅ ensures that F is meager.
III. A new approach
The first classical proof of the E0 dichotomy theorem
Sketch of the proof
Set G = E c and H = E \ ∆(X ).
One can construct a Borel reduction of E to ∆(2ω ) from Borel functions φ : X → ω and ψ : X → 2ω for which φ × ψ is a
coloring of G and ψ is a homomorphism from H to ∆(2ω ).
By the dichotomy for pairs, we can assume that there is a
continuous homomorphism φ from (G0even , H0odd ) to (E c , E ).
Then F = (φ × φ)−1 (E ) has the Baire property.
The fact that G0even ∩ F = ∅ ensures that F is meager.
The fact that H0odd ⊆ F then allows us to construct a continuous homomorphism ψ from the triple (∆(2ω )c , E0c , E0 ) to the
triple ((φ × φ)−1 (∆(X ))c , F c , F ).
III. A new approach
The first classical proof of the E0 dichotomy theorem
Sketch of the proof
Set G = E c and H = E \ ∆(X ).
One can construct a Borel reduction of E to ∆(2ω ) from Borel functions φ : X → ω and ψ : X → 2ω for which φ × ψ is a
coloring of G and ψ is a homomorphism from H to ∆(2ω ).
By the dichotomy for pairs, we can assume that there is a
continuous homomorphism φ from (G0even , H0odd ) to (E c , E ).
Then F = (φ × φ)−1 (E ) has the Baire property.
The fact that G0even ∩ F = ∅ ensures that F is meager.
The fact that H0odd ⊆ F then allows us to construct a continuous homomorphism ψ from the triple (∆(2ω )c , E0c , E0 ) to the
triple ((φ × φ)−1 (∆(X ))c , F c , F ).
The function φ ◦ ψ is as desired.
Part IV
Generalizations
IV. Generalizations
IV. Generalizations
Definition
A Hausdorff space X is κ-Souslin if it is the continuous image of a
closed subset of κω .
IV. Generalizations
Definition
A Hausdorff space X is κ-Souslin if it is the continuous image of a
closed subset of κω .
Remark
Much work has been done generalizing descriptive set-theoretic dichotomy theorems from Borel sets to projective sets and beyond.
IV. Generalizations
IV. Generalizations
Remark
This is typically accomplished under AD by finding generalizations
of these theorems to κ-Souslin sets.
IV. Generalizations
Remark
This is typically accomplished under AD by finding generalizations
of these theorems to κ-Souslin sets.
Remark
Arguments relying upon the effective theory rarely go through for
κ-Souslin sets.
IV. Generalizations
IV. Generalizations
Important observation
A simplification(!) of the classical proof of the G0 dichotomy can
be used to establish a weak generalization to κ-Souslin graphs.
IV. Generalizations
Important observation
A simplification(!) of the classical proof of the G0 dichotomy can
be used to establish a weak generalization to κ-Souslin graphs.
Theorem
Suppose that X is a Hausdorff space and G is a κ-Souslin graph on
X . Then at least one of the following holds:
There is a κ-coloring of G.
There is a continuous homomorphism from G0 to G.
IV. Generalizations
IV. Generalizations
Definition
A set B ⊆ X is ω-universally Baire if for every continuous function
π : 2ω → X , the set π −1 (B) has the Baire property.
IV. Generalizations
Definition
A set B ⊆ X is ω-universally Baire if for every continuous function
π : 2ω → X , the set π −1 (B) has the Baire property.
Important observation
By combining the κ-Souslin analog of the G0 dichotomy with the
classical proofs mentioned earlier, one can prove analogs of many
theorems for κ-Souslin structures which are ω-universally Baire.
IV. Generalizations
IV. Generalizations
Theorem
Suppose that κ is an infinite aleph, X is a Hausdorff space, and E
is a co-κ-Souslin equivalence relation on X which is ω-universally
Baire. Then at least one of the following holds:
The set X /E has cardinality at most κ.
There is a continuous embedding of ∆(2ω ) into E .
IV. Generalizations
Theorem
Suppose that κ is an infinite aleph, X is a Hausdorff space, and E
is a co-κ-Souslin equivalence relation on X which is ω-universally
Baire. Then at least one of the following holds:
The set X /E has cardinality at most κ.
There is a continuous embedding of ∆(2ω ) into E .
Remark
This yields the results of Silver and Burgess, as well as the Harrington-Shelah generalization to absolutely ∆12 equivalence relations.
IV. Generalizations
IV. Generalizations
Theorem
Suppose that κ is an infinite aleph, X is a Hausdorff space, and E
is a bi-κ-Souslin equivalence relation on X which is ω-universally
Baire. Then at least one of the following holds:
There is a reduction of E to ∆(2κ ).
There is a continuous embedding of E0 into E .
IV. Generalizations
Theorem
Suppose that κ is an infinite aleph, X is a Hausdorff space, and E
is a bi-κ-Souslin equivalence relation on X which is ω-universally
Baire. Then at least one of the following holds:
There is a reduction of E to ∆(2κ ).
There is a continuous embedding of E0 into E .
Remark
This was previously proven by both Ditzen and Foreman-Magidor
under ZF + AD.
IV. Generalizations
IV. Generalizations
Remark
Ideally, we would like to have the strengthening of this theorem in
which the reduction is κ+ -Borel.
IV. Generalizations
Remark
Ideally, we would like to have the strengthening of this theorem in
which the reduction is κ+ -Borel.
Important observation (Hjorth)
By employing methods from forcing and infinitary logic, one can
establish the analogous generalization of the Kechris-Solecki-Todorcevic theorem.
IV. Generalizations
IV. Generalizations
Theorem (Hjorth)
Suppose that X is a Hausdorff space and G is a κ-Souslin graph on
X . Then at least one of the following holds:
There is a κ+ -Borel κ-coloring of G.
There is a continuous homomorphism from G0 to G.
IV. Generalizations
Theorem (Hjorth)
Suppose that X is a Hausdorff space and G is a κ-Souslin graph on
X . Then at least one of the following holds:
There is a κ+ -Borel κ-coloring of G.
There is a continuous homomorphism from G0 to G.
Remark
This simultaneously generalizes the Kechris-Solecki-Todorcevic
theorem and strengthens its κ-Souslin analog mentioned earlier.
IV. Generalizations
IV. Generalizations
Remark
This result appears to give the desired strengthening of the Harrington-Kechris-Louveau theorem.
IV. Generalizations
Remark
This result appears to give the desired strengthening of the Harrington-Kechris-Louveau theorem.
Theorem
Suppose that κ is an infinite aleph, X is a Hausdorff space, and E
is a bi-κ-Souslin equivalence relation on X which is ω-universally
Baire. Then at least one of the following holds:
There is a κ+ -Borel reduction of E to ∆(2κ ).
There is a continuous embedding of E0 into E .
IV. Generalizations
IV. Generalizations
Remark
One can use Hjorth’s result to generalize many other theorems.
IV. Generalizations
Remark
One can use Hjorth’s result to generalize many other theorems.
Theorem
Suppose that κ is an infinite aleph, X and Y are Hausdorff spaces,
and R ⊆ X × Y is κ-Souslin. Then one of the following holds:
There are partial functions fα ··· X →SY , whose graphs are κ+ Borel subsets of R, such that R = α∈κ graph(fα ).
For some x ∈ X , there is a continuous injection of 2ω into Rx .
IV. Generalizations
IV. Generalizations
Remark
Unfortunately, the fact that Hjorth’s proof is non-classical means
that proofs using it are also non-classical.
IV. Generalizations
Remark
Unfortunately, the fact that Hjorth’s proof is non-classical means
that proofs using it are also non-classical.
Question
Is there a classical proof of Hjorth’s theorem?
IV. Generalizations
IV. Generalizations
Definition
A Hausdorff space X is weakly κ-Souslin if it is the continuous image of a κ+ -Borel subset of κω .
IV. Generalizations
Definition
A Hausdorff space X is weakly κ-Souslin if it is the continuous image of a κ+ -Borel subset of κω .
Question
Suppose that X is a Hausdorff space and A, B ⊆ X are disjoint
weakly κ-Souslin sets. Is there a κ+ -Borel set C ⊆ X with A ⊆ C
and B ∩ C = ∅? If so, is there a classical proof?
IV. Generalizations
Definition
A Hausdorff space X is weakly κ-Souslin if it is the continuous image of a κ+ -Borel subset of κω .
Question
Suppose that X is a Hausdorff space and A, B ⊆ X are disjoint
weakly κ-Souslin sets. Is there a κ+ -Borel set C ⊆ X with A ⊆ C
and B ∩ C = ∅? If so, is there a classical proof?
Remark
A positive answer should give a classical proof of Hjorth’s theorem.
The consistent failure of the separation theorem would provide evidence for the necessity of Hjorth’s approach.