Potential Wadge classes

Lecomte, Dominique

doi:10.1090/s0065-9266-2012-00658-7

Cited by 3 publications

(32 citation statements)

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“…If

boldΓ \subseteq Δ_{2}^{0}

, we can find disjoint analytic locally countable relations

A, B

on 2 ω such that A is not separable from B by a

pot (boldΓ)

set, as we shall see. This shows that, in order to get partial reductions with injectivity, we have to use examples different from those in , so that we prove the following.…”

Section: The Classes Normaldηfalse(boldς10false) and Trued˘ηfalse(bolmentioning

confidence: 88%

“…Proof This result is essentially proved in . However, the formula for

F_{η, ξ}^{ε}

is more concrete here, since the more general and abstract case of Wadge classes is considered in . So we give some details.…”

Section: The Classes Normaldηfalse(boldς10false) and Trued˘ηfalse(bolmentioning

confidence: 95%

“…Our motivating result is Theorem (cf. ). Definition We say that a class

Γ

of subsets of zero‐dimensional Polish spaces is a Wadge class of Borel sets if there is a Borel subset B of

ω^{ω}

such that for any zero‐dimensional Polish space X , and for any

A \subseteq X

, A is in

Γ

if and only if there is

f : X \to ω^{ω}

continuous such that

A = f^{- 1} false(B false)

.…”

Section: Introductionmentioning

confidence: 97%

“…Debs proved that this is the case if

Γ

is a non self‐dual Borel class of rank at least three (i.e., a class

{boldΣ}_{ξ}^{0}

{boldΠ}_{ξ}^{0}

with

ξ \geq 3

). As mentioned in , there is also an injectivity result for the non self‐dual Wadge classes of Borel sets of level at least three. [, Theorem 2.16], [, Theorem 15], and [, Theorems 7.1 and 7.8] show that we cannot have f and g injective if (b) holds and

Γ

is a non self‐dual Borel class of rank one or two, or the class of clopen sets, because of cycle problems again.…”

Section: Introductionmentioning

confidence: 99%

“…The notion of potential complexity is a natural invariant for ⩽ B : if

E \leq_{normalB} F

and

F \in pot (boldΓ)

, then

E \in pot (boldΓ)

too. However, as shown in , ⩽ B is not the right notion of comparison to study potential complexity, in the general context, because of cycle problems. A good notion of comparison is as follows.…”

Section: Introductionmentioning

confidence: 99%

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Injective tests of low complexity in the plane

Lecomte

Zamora

2019

Mathematical Logic Qtrly

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“…If

boldΓ \subseteq Δ_{2}^{0}

, we can find disjoint analytic locally countable relations

A, B

on 2 ω such that A is not separable from B by a

pot (boldΓ)

set, as we shall see. This shows that, in order to get partial reductions with injectivity, we have to use examples different from those in , so that we prove the following.…”

Section: The Classes Normaldηfalse(boldς10false) and Trued˘ηfalse(bolmentioning

confidence: 88%

“…Proof This result is essentially proved in . However, the formula for

F_{η, ξ}^{ε}

is more concrete here, since the more general and abstract case of Wadge classes is considered in . So we give some details.…”

Section: The Classes Normaldηfalse(boldς10false) and Trued˘ηfalse(bolmentioning

confidence: 95%

“…Our motivating result is Theorem (cf. ). Definition We say that a class

Γ

of subsets of zero‐dimensional Polish spaces is a Wadge class of Borel sets if there is a Borel subset B of

ω^{ω}

such that for any zero‐dimensional Polish space X , and for any

A \subseteq X

, A is in

Γ

if and only if there is

f : X \to ω^{ω}

continuous such that

A = f^{- 1} false(B false)

.…”

Section: Introductionmentioning

confidence: 97%

“…Debs proved that this is the case if

Γ

is a non self‐dual Borel class of rank at least three (i.e., a class

{boldΣ}_{ξ}^{0}

{boldΠ}_{ξ}^{0}

with

ξ \geq 3

Γ

is a non self‐dual Borel class of rank one or two, or the class of clopen sets, because of cycle problems again.…”

Section: Introductionmentioning

confidence: 99%

“…The notion of potential complexity is a natural invariant for ⩽ B : if

E \leq_{normalB} F

and

F \in pot (boldΓ)

, then

E \in pot (boldΓ)

Section: Introductionmentioning

confidence: 99%

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Injective tests of low complexity in the plane

Lecomte

Zamora

2019

Mathematical Logic Qtrly

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On the complexity of Borel equivalence relations with some countability property

Lecomte

2019

Trans. Amer. Math. Soc.

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We study the class of Borel equivalence relations under continuous reducibility. In particular, we characterize when a Borel equivalence relation with countable equivalence classes is Σ 0 ξ (or Π 0 ξ ). We characterize when all the equivalence classes of such a relation are Σ 0 ξ (or Π 0 ξ ). We prove analogous results for the Borel equivalence relations with countably many equivalence classes. We also completely solve these two problems for the first two ranks. In order to do this, we prove some extensions of the Louveau-Saint Raymond theorem which itself generalized the Hurewicz theorem characterizing when a Borel subset of a Polish space is G δ .

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