F F F F 5FF AnewspacefromaprojectiveFra¨ıss´elimit

A new space from a projective Fra¨ıss´e limit

joint work (in progress) with G. Basso

or

5F

F^rom F^{ans to} F^{ences via} F^ra¨ıss´e F^amilies

A new space from a projective Fra¨ıss´e limit

joint work (in progress) with G. Basso or

5F

F^rom F^{ans to} F^{ences via} F^ra¨ıss´e F^amilies

A new space from a projective Fra¨ıss´e limit

joint work (in progress) with G. Basso or

5F

F^rom F^{ans to} F^{ences via} F^ra¨ıss´e F^amilies

Introduction

Main basic notions:

I topological structures

I projective Fra¨ıss´e families

I projective Fra¨ıss´e limits

These have been introduced by T. Irwin and S. Solecki (2006).

Introduction

Main basic notions:

I topological structures

I projective Fra¨ıss´e families

I projective Fra¨ıss´e limits

These have been introduced by T. Irwin and S. Solecki (2006).

Topological structures

Let L be a first-order language.

I An L-structure A is a topological L-structure if:

I A carries a compact (Hausdorff), zero-dimensional, second countable topology

I the interpretations of the relation symbols are closed sets

I the interpretations of the function symbols are continuous functions

Topological structures

Let L be a first-order language.

I An L-structure A is a topological L-structure if:

I A carries a compact (Hausdorff), zero-dimensional, second countable topology

I the interpretations of the relation symbols are closed sets

I the interpretations of the function symbols are continuous functions

Topological structures

Let L be a first-order language.

I An L-structure A is a topological L-structure if:

I A carries a compact (Hausdorff), zero-dimensional, second countable topology

I the interpretations of the relation symbols are closed sets

I the interpretations of the function symbols are continuous functions

Epimorphisms

I A continuous surjection ϕ : A → B between topological L-structures A, B is an epimorphism if

I f^B= (ϕ(a1), . . . , ϕ(an)) = ϕf^A(a1, . . . , an), for every n-ary function symbol f ∈ L

I c^B= ϕ(c^A) for every constant symbol c ∈ L

I r^B= ϕ × . . . × ϕ

| {z }

n times

(r^A) for every n-ary relation symbol r ∈ L; that is

r^B(b1, . . . , bn) iff

∃a1, . . . , an∈ A (ϕ(a1) = b1∧ . . . ∧ ϕ(an) = bn∧ r^A(a1, . . . , an))

I If there is an epimorphism A → B, write B E A

I An isomorphism between topological L-structures is a bijective epimorphism. So, it is

I an isomorphism between L-structures

I a homeomorphism between their supports

I An epimorphism ϕ : A → B refines a covering U of A if the preimage of any element of B is contained in some member of U .

Epimorphisms

I A continuous surjection ϕ : A → B between topological L-structures A, B is an epimorphism if

I f^B = (ϕ(a1), . . . , ϕ(an)) = ϕf^A(a1, . . . , an), for every n-ary function symbol f ∈ L

I c^B= ϕ(c^A) for every constant symbol c ∈ L

I r^B= ϕ × . . . × ϕ

| {z }

n times

(r^A) for every n-ary relation symbol r ∈ L; that is

r^B(b1, . . . , bn) iff

∃a1, . . . , an∈ A (ϕ(a1) = b1∧ . . . ∧ ϕ(an) = bn∧ r^A(a1, . . . , an))

I If there is an epimorphism A → B, write B E A

I An isomorphism between topological L-structures is a bijective epimorphism. So, it is

I an isomorphism between L-structures

I a homeomorphism between their supports

I An epimorphism ϕ : A → B refines a covering U of A if the preimage of any element of B is contained in some member of U .

Epimorphisms

I A continuous surjection ϕ : A → B between topological L-structures A, B is an epimorphism if

I f^B = (ϕ(a1), . . . , ϕ(an)) = ϕf^A(a1, . . . , an), for every n-ary function symbol f ∈ L

I c^B= ϕ(c^A) for every constant symbol c ∈ L

I r^B = ϕ × . . . × ϕ

| {z }

n times

(r^A) for every n-ary relation symbol r ∈ L;

that is

r^B(b1, . . . , bn) iff

∃a1, . . . , an∈ A (ϕ(a1) = b1∧ . . . ∧ ϕ(an) = bn∧ r^A(a1, . . . , an))

I If there is an epimorphism A → B, write B E A

I An isomorphism between topological L-structures is a bijective epimorphism. So, it is

I an isomorphism between L-structures

I a homeomorphism between their supports

I An epimorphism ϕ : A → B refines a covering U of A if the preimage of any element of B is contained in some member of U .

Epimorphisms

I A continuous surjection ϕ : A → B between topological L-structures A, B is an epimorphism if

I f^B = (ϕ(a1), . . . , ϕ(an)) = ϕf^A(a1, . . . , an), for every n-ary function symbol f ∈ L

I c^B= ϕ(c^A) for every constant symbol c ∈ L

I r^B = ϕ × . . . × ϕ

| {z }

n times

(r^A) for every n-ary relation symbol r ∈ L; that is

r^B(b1, . . . , bn) iff

∃a1, . . . , an∈ A (ϕ(a1) = b1∧ . . . ∧ ϕ(an) = bn∧ r^A(a1, . . . , an))

I If there is an epimorphism A → B, write B E A

I An isomorphism between topological L-structures is a bijective epimorphism. So, it is

I an isomorphism between L-structures

I a homeomorphism between their supports

I An epimorphism ϕ : A → B refines a covering U of A if the preimage of any element of B is contained in some member of U .

Epimorphisms

I A continuous surjection ϕ : A → B between topological L-structures A, B is an epimorphism if

I f^B = (ϕ(a1), . . . , ϕ(an)) = ϕf^A(a1, . . . , an), for every n-ary function symbol f ∈ L

I c^B= ϕ(c^A) for every constant symbol c ∈ L

I r^B = ϕ × . . . × ϕ

| {z }

n times

(r^A) for every n-ary relation symbol r ∈ L; that is

r^B(b1, . . . , bn) iff

∃a1, . . . , an∈ A (ϕ(a1) = b1∧ . . . ∧ ϕ(an) = bn∧ r^A(a1, . . . , an))

I If there is an epimorphism A → B, write B E A

I An isomorphism between topological L-structures is a bijective epimorphism. So, it is

I an isomorphism between L-structures

I a homeomorphism between their supports

I An epimorphism ϕ : A → B refines a covering U of A if the preimage of any element of B is contained in some member of U .

Epimorphisms

I A continuous surjection ϕ : A → B between topological L-structures A, B is an epimorphism if

I f^B = (ϕ(a1), . . . , ϕ(an)) = ϕf^A(a1, . . . , an), for every n-ary function symbol f ∈ L

I c^B= ϕ(c^A) for every constant symbol c ∈ L

I r^B = ϕ × . . . × ϕ

| {z }

n times

(r^A) for every n-ary relation symbol r ∈ L; that is

r^B(b1, . . . , bn) iff

∃a1, . . . , an∈ A (ϕ(a1) = b1∧ . . . ∧ ϕ(an) = bn∧ r^A(a1, . . . , an))

I If there is an epimorphism A → B, write B E A

I An isomorphism between topological L-structures is a bijective epimorphism.

So, it is

I an isomorphism between L-structures

I a homeomorphism between their supports

I An epimorphism ϕ : A → B refines a covering U of A if the preimage of any element of B is contained in some member of U .

Epimorphisms

I A continuous surjection ϕ : A → B between topological L-structures A, B is an epimorphism if

I f^B = (ϕ(a1), . . . , ϕ(an)) = ϕf^A(a1, . . . , an), for every n-ary function symbol f ∈ L

I c^B= ϕ(c^A) for every constant symbol c ∈ L

I r^B = ϕ × . . . × ϕ

| {z }

n times

(r^A) for every n-ary relation symbol r ∈ L; that is

r^B(b1, . . . , bn) iff

∃a1, . . . , an∈ A (ϕ(a1) = b1∧ . . . ∧ ϕ(an) = bn∧ r^A(a1, . . . , an))

I If there is an epimorphism A → B, write B E A

I An isomorphism between topological L-structures is a bijective epimorphism. So, it is

I an isomorphism between L-structures

I a homeomorphism between their supports

I An epimorphism ϕ : A → B refines a covering U of A if the preimage of any element of B is contained in some member of U .

Epimorphisms

I A continuous surjection ϕ : A → B between topological L-structures A, B is an epimorphism if

I f^B = (ϕ(a1), . . . , ϕ(an)) = ϕf^A(a1, . . . , an), for every n-ary function symbol f ∈ L

I c^B= ϕ(c^A) for every constant symbol c ∈ L

I r^B = ϕ × . . . × ϕ

| {z }

n times

(r^A) for every n-ary relation symbol r ∈ L; that is

r^B(b1, . . . , bn) iff

∃a1, . . . , an∈ A (ϕ(a1) = b1∧ . . . ∧ ϕ(an) = bn∧ r^A(a1, . . . , an))

I If there is an epimorphism A → B, write B E A

I An isomorphism between topological L-structures is a bijective epimorphism. So, it is

I an isomorphism between L-structures

I a homeomorphism between their supports

I An epimorphism ϕ : A → B refines a covering U of A if

the preimage of any element of B is contained in some member of U .

Epimorphisms

I A continuous surjection ϕ : A → B between topological L-structures A, B is an epimorphism if

I f^B = (ϕ(a1), . . . , ϕ(an)) = ϕf^A(a1, . . . , an), for every n-ary function symbol f ∈ L

I c^B= ϕ(c^A) for every constant symbol c ∈ L

I r^B = ϕ × . . . × ϕ

| {z }

n times

(r^A) for every n-ary relation symbol r ∈ L; that is

r^B(b1, . . . , bn) iff

∃a1, . . . , an∈ A (ϕ(a1) = b1∧ . . . ∧ ϕ(an) = bn∧ r^A(a1, . . . , an))

I If there is an epimorphism A → B, write B E A

I An isomorphism between topological L-structures is a bijective epimorphism. So, it is

I an isomorphism between L-structures

I a homeomorphism between their supports

I An epimorphism ϕ : A → B refines a covering U of A if the preimage of any element of B is contained in some member of U .

Projective Fra¨ıss´e families

Let F be a family of topological L-structures.

I F is a projective Fra¨ıss´e family if the following properties hold: JPP (joint projection property) ∀A, B ∈ F ∃C ∈ F (A E C ∧ B E C )

AP (amalgamation property) for every A, B, C ∈ F and epimorphisms ϕ1: B → A, ϕ2: C → A

there exist D ∈ F and epimorphisms

ψ1: D → B, ψ2: D → C such that

ϕ1ψ1= ϕ2ψ2

Projective Fra¨ıss´e families

Let F be a family of topological L-structures.

I F is a projective Fra¨ıss´e family if

the following properties hold: JPP (joint projection property) ∀A, B ∈ F ∃C ∈ F (A E C ∧ B E C )

AP (amalgamation property) for every A, B, C ∈ F and epimorphisms ϕ1: B → A, ϕ2: C → A

there exist D ∈ F and epimorphisms

ψ1: D → B, ψ2: D → C such that

ϕ1ψ1= ϕ2ψ2

Projective Fra¨ıss´e families

Let F be a family of topological L-structures.

I F is a projective Fra¨ıss´e family if the following properties hold:

JPP (joint projection property)

∀A, B ∈ F ∃C ∈ F (A E C ∧ B E C ) AP (amalgamation property) for every A, B, C ∈ F and epimorphisms

ϕ1: B → A, ϕ2: C → A there exist D ∈ F and epimorphisms

ψ1: D → B, ψ2: D → C such that

ϕ1ψ1= ϕ2ψ2

Projective Fra¨ıss´e families

Let F be a family of topological L-structures.

I F is a projective Fra¨ıss´e family if the following properties hold:

JPP (joint projection property)∀A, B ∈ F ∃C ∈ F (A E C ∧ B E C )

AP (amalgamation property) for every A, B, C ∈ F and epimorphisms ϕ1: B → A, ϕ2: C → A

there exist D ∈ F and epimorphisms

ψ1: D → B, ψ2: D → C such that

ϕ1ψ1= ϕ2ψ2

Projective Fra¨ıss´e families

Let F be a family of topological L-structures.

I F is a projective Fra¨ıss´e family if the following properties hold:

JPP (joint projection property)∀A, B ∈ F ∃C ∈ F (A E C ∧ B E C ) AP (amalgamation property)

for every A, B, C ∈ F and epimorphisms ϕ1: B → A, ϕ2: C → A

there exist D ∈ F and epimorphisms

ψ1: D → B, ψ2: D → C such that

ϕ1ψ1= ϕ2ψ2

Projective Fra¨ıss´e families

Let F be a family of topological L-structures.

I F is a projective Fra¨ıss´e family if the following properties hold:

JPP (joint projection property)∀A, B ∈ F ∃C ∈ F (A E C ∧ B E C ) AP (amalgamation property) for every A, B, C ∈ F and epimorphisms

ϕ1: B → A, ϕ2: C → A

there exist D ∈ F and epimorphisms

ψ1: D → B, ψ2: D → C such that

ϕ1ψ1= ϕ2ψ2

Projective Fra¨ıss´e families

Let F be a family of topological L-structures.

I F is a projective Fra¨ıss´e family if the following properties hold:

JPP (joint projection property)∀A, B ∈ F ∃C ∈ F (A E C ∧ B E C ) AP (amalgamation property) for every A, B, C ∈ F and epimorphisms

ϕ1: B → A, ϕ2: C → A there exist D ∈ F and epimorphisms

ψ1: D → B, ψ2: D → C such that

ϕ1ψ1= ϕ2ψ2

Projective Fra¨ıss´e limits

Let

I F a family of topological L-structures

I P a topological L-structure

P is a projective Fra¨ıss´e limit of F if the following properties hold: L1 (projective universality) ∀A ∈ F A E P

L2’ for every clopen partition U of P there exist A ∈ F and an epimorphism ϕ : P → A refining U

L3 (projective ultrahomogeneity) for every A ∈ F and epimorphisms ϕ1, ϕ2: P → A

there exists an isomorphism

ψ : P → P such that

ϕ2= ϕ1ψ

Projective Fra¨ıss´e limits

Let

I F a family of topological L-structures

I P a topological L-structure

P is a projective Fra¨ıss´e limit of F if the following properties hold: L1 (projective universality) ∀A ∈ F A E P

L2’ for every clopen partition U of P there exist A ∈ F and an epimorphism ϕ : P → A refining U

L3 (projective ultrahomogeneity) for every A ∈ F and epimorphisms ϕ1, ϕ2: P → A

there exists an isomorphism

ψ : P → P such that

ϕ2= ϕ1ψ

Projective Fra¨ıss´e limits

Let

I F a family of topological L-structures

I P a topological L-structure

P is a projective Fra¨ıss´e limit of F if the following properties hold:

L1 (projective universality) ∀A ∈ F A E P

L2’ for every clopen partition U of P there exist A ∈ F and an epimorphism ϕ : P → A refining U

L3 (projective ultrahomogeneity) for every A ∈ F and epimorphisms ϕ1, ϕ2: P → A

there exists an isomorphism

ψ : P → P such that

ϕ2= ϕ1ψ

Projective Fra¨ıss´e limits

Let

I F a family of topological L-structures

I P a topological L-structure

P is a projective Fra¨ıss´e limit of F if the following properties hold:

L1 (projective universality)

∀A ∈ F A E P

L2’ for every clopen partition U of P there exist A ∈ F and an epimorphism ϕ : P → A refining U

L3 (projective ultrahomogeneity) for every A ∈ F and epimorphisms ϕ1, ϕ2: P → A

there exists an isomorphism

ψ : P → P such that

ϕ2= ϕ1ψ

Projective Fra¨ıss´e limits

Let

I F a family of topological L-structures

I P a topological L-structure

P is a projective Fra¨ıss´e limit of F if the following properties hold:

L1 (projective universality)∀A ∈ F A E P

L2’ for every clopen partition U of P there exist A ∈ F and an epimorphism ϕ : P → A refining U

L3 (projective ultrahomogeneity) for every A ∈ F and epimorphisms ϕ1, ϕ2: P → A

there exists an isomorphism

ψ : P → P such that

ϕ2= ϕ1ψ

Projective Fra¨ıss´e limits

Let

I F a family of topological L-structures

I P a topological L-structure

P is a projective Fra¨ıss´e limit of F if the following properties hold:

L1 (projective universality)∀A ∈ F A E P L2’ for every clopen partition U of P

there exist A ∈ F and an epimorphism ϕ : P → A refining U

L3 (projective ultrahomogeneity) for every A ∈ F and epimorphisms ϕ1, ϕ2: P → A

there exists an isomorphism

ψ : P → P such that

ϕ2= ϕ1ψ

Projective Fra¨ıss´e limits

Let

I F a family of topological L-structures

I P a topological L-structure

P is a projective Fra¨ıss´e limit of F if the following properties hold:

L1 (projective universality)∀A ∈ F A E P

L2’ for every clopen partition U of P there exist A ∈ F and an epimorphism ϕ : P → A refining U

L3 (projective ultrahomogeneity) for every A ∈ F and epimorphisms ϕ1, ϕ2: P → A

there exists an isomorphism

ψ : P → P such that

ϕ2= ϕ1ψ

Projective Fra¨ıss´e limits

Let

I F a family of topological L-structures

I P a topological L-structure

P is a projective Fra¨ıss´e limit of F if the following properties hold:

L1 (projective universality)∀A ∈ F A E P

L2’ for every clopen partition U of P there exist A ∈ F and an epimorphism ϕ : P → A refining U

L3 (projective ultrahomogeneity)

for every A ∈ F and epimorphisms ϕ1, ϕ2: P → A

there exists an isomorphism

ψ : P → P such that

ϕ2= ϕ1ψ

Projective Fra¨ıss´e limits

Let

I F a family of topological L-structures

I P a topological L-structure

P is a projective Fra¨ıss´e limit of F if the following properties hold:

L1 (projective universality)∀A ∈ F A E P

L2’ for every clopen partition U of P there exist A ∈ F and an epimorphism ϕ : P → A refining U

L3 (projective ultrahomogeneity) for every A ∈ F and epimorphisms ϕ1, ϕ2: P → A

there exists an isomorphism

ψ : P → P such that

ϕ2= ϕ1ψ

Projective Fra¨ıss´e limits

Let

I F a family of topological L-structures

I P a topological L-structure

P is a projective Fra¨ıss´e limit of F if the following properties hold:

L1 (projective universality)∀A ∈ F A E P

L2’ for every clopen partition U of P there exist A ∈ F and an epimorphism ϕ : P → A refining U

L3 (projective ultrahomogeneity) for every A ∈ F and epimorphisms ϕ1, ϕ2: P → A

there exists an isomorphism

ψ : P → P such that

ϕ2= ϕ1ψ

Fundamental sequences

Let F be a family of topological L-structures.

A fundamental sequence (Dn, πn) is a sequence of elements Dn∈ F together with epimorphisms πn: Dn+1→ Dn such that, denoting

π_n^m= πn· . . . · πm−1: Dm→ Dn, for m ≥ n the following properties hold:

I ∀A ∈ F ∃n ∈ N A E Dn

I for every n, every A, B ∈ F , every epimorphisms ϕ₁: B → A, ϕ₂: D_n→ A there exist m ≥ n and an epimorphism

ψ : Dm→ B such that

ϕ1ψ = ϕ2π_n^m

Fundamental sequences

Let F be a family of topological L-structures. A fundamental sequence (Dn, πn) is a sequence of elements Dn∈ F together with epimorphisms πn: Dn+1→ Dnsuch that,

denoting

π_n^m= πn· . . . · πm−1: Dm→ Dn, for m ≥ n the following properties hold:

I ∀A ∈ F ∃n ∈ N A E Dn

I for every n, every A, B ∈ F , every epimorphisms ϕ₁: B → A, ϕ₂: D_n→ A there exist m ≥ n and an epimorphism

ψ : Dm→ B such that

ϕ1ψ = ϕ2π_n^m

Fundamental sequences

Let F be a family of topological L-structures. A fundamental sequence (Dn, πn) is a sequence of elements Dn∈ F together with epimorphisms πn: Dn+1→ Dnsuch that, denoting

π_n^m= πn· . . . · πm−1: Dm→ Dn, for m ≥ n

the following properties hold:

I ∀A ∈ F ∃n ∈ N A E Dn

I for every n, every A, B ∈ F , every epimorphisms ϕ₁: B → A, ϕ₂: D_n→ A there exist m ≥ n and an epimorphism

ψ : Dm→ B such that

ϕ1ψ = ϕ2π_n^m

Fundamental sequences

Let F be a family of topological L-structures. A fundamental sequence (Dn, πn) is a sequence of elements Dn∈ F together with epimorphisms πn: Dn+1→ Dnsuch that, denoting

π_n^m= πn· . . . · πm−1: Dm→ Dn, for m ≥ n the following properties hold:

I ∀A ∈ F ∃n ∈ N A E Dn

I for every n, every A, B ∈ F , every epimorphisms ϕ₁: B → A, ϕ₂: D_n→ A there exist m ≥ n and an epimorphism

ψ : Dm→ B such that

ϕ1ψ = ϕ2π_n^m

Fundamental sequences

Let F be a family of topological L-structures. A fundamental sequence (Dn, πn) is a sequence of elements Dn∈ F together with epimorphisms πn: Dn+1→ Dnsuch that, denoting

π_n^m= πn· . . . · πm−1: Dm→ Dn, for m ≥ n the following properties hold:

I ∀A ∈ F ∃n ∈ N A E Dn

I for every n, every A, B ∈ F , every epimorphisms ϕ₁: B → A, ϕ₂: D_n→ A

there exist m ≥ n and an epimorphism ψ : Dm→ B such that

ϕ1ψ = ϕ2π_n^m

Fundamental sequences

Let F be a family of topological L-structures. A fundamental sequence (Dn, πn) is a sequence of elements Dn∈ F together with epimorphisms πn: Dn+1→ Dnsuch that, denoting

π_n^m= πn· . . . · πm−1: Dm→ Dn, for m ≥ n the following properties hold:

I ∀A ∈ F ∃n ∈ N A E Dn

I for every n, every A, B ∈ F , every epimorphisms ϕ₁: B → A, ϕ₂: D_n→ A there exist m ≥ n and an epimorphism

ψ : Dm→ B such that

ϕ1ψ = ϕ2π_n^m

A basic result

Theorem (essentially: Irwin, Solecki - 2006)

Let F be a non-empty, at most countable family of finite topological L-structures. Then the following are equivalent:

1. F is a projective Fra¨ıss´e family 2. F has a projective Fra¨ıss´e limit 3. F has a fundamental sequence

Moreover, in this case

I the projective Fra¨ıss´e limit of F is unique up to isomorphism

I the projective Fra¨ıss´e limits of F and of its fundamental sequence coincide

I a projective Fra¨ıss´e limit for both is the inverse limit of the fundamental sequence

A basic result

Theorem (essentially: Irwin, Solecki - 2006)

Let F be a non-empty, at most countable family of finite topological L-structures. Then the following are equivalent:

1. F is a projective Fra¨ıss´e family 2. F has a projective Fra¨ıss´e limit 3. F has a fundamental sequence Moreover, in this case

I the projective Fra¨ıss´e limit of F is unique up to isomorphism

I the projective Fra¨ıss´e limits of F and of its fundamental sequence coincide

I a projective Fra¨ıss´e limit for both is the inverse limit of the fundamental sequence

A basic result

Theorem (essentially: Irwin, Solecki - 2006)

Let F be a non-empty, at most countable family of finite topological L-structures. Then the following are equivalent:

1. F is a projective Fra¨ıss´e family 2. F has a projective Fra¨ıss´e limit 3. F has a fundamental sequence Moreover, in this case

I the projective Fra¨ıss´e limit of F is unique up to isomorphism

I the projective Fra¨ıss´e limits of F and of its fundamental sequence coincide

I a projective Fra¨ıss´e limit for both is the inverse limit of the fundamental sequence

A basic result

Theorem (essentially: Irwin, Solecki - 2006)

Let F be a non-empty, at most countable family of finite topological L-structures. Then the following are equivalent:

1. F is a projective Fra¨ıss´e family 2. F has a projective Fra¨ıss´e limit 3. F has a fundamental sequence Moreover, in this case

I the projective Fra¨ıss´e limit of F is unique up to isomorphism

I the projective Fra¨ıss´e limits of F and of its fundamental sequence coincide

I a projective Fra¨ıss´e limit for both is the inverse limit of the fundamental sequence

Prespaces

Denote by LR a first-order language with a distinguished binary relation symbol R, that is

LR = {R, . . .}

I A prespace is any topological L-structure A such that R^A is an equivalence relation

I If (A, R^A, . . .) is a prespace, then it is a prespace of every compact metrisable space homeomorphic to A/R^A

Prespaces

Denote by LR a first-order language with a distinguished binary relation symbol R,

that is

LR = {R, . . .}

I A prespace is any topological L-structure A such that R^A is an equivalence relation

I If (A, R^A, . . .) is a prespace, then it is a prespace of every compact metrisable space homeomorphic to A/R^A

Prespaces

Denote by LR a first-order language with a distinguished binary relation symbol R, that is

LR = {R, . . .}

I A prespace is any topological L-structure A such that R^A is an equivalence relation

I If (A, R^A, . . .) is a prespace, then it is a prespace of every compact metrisable space homeomorphic to A/R^A

Prespaces

Denote by LR a first-order language with a distinguished binary relation symbol R, that is

LR = {R, . . .}

I A prespace is any topological L-structure A such that R^A is an equivalence relation

I If (A, R^A, . . .) is a prespace, then it is a prespace of every compact metrisable space homeomorphic to A/R^A

Prespaces

Denote by LR a first-order language with a distinguished binary relation symbol R, that is

LR = {R, . . .}

I A prespace is any topological L-structure A such that R^A is an equivalence relation

I If (A, R^A, . . .) is a prespace, then it is a prespace of every compact metrisable space homeomorphic to A/R^A

General goal

I Start with a non-empty, at most countable, projective Fra¨ıss´e family F of finite topological LR-structures

I Compute its limit P

I Determine if P is a prespace; that is, determine if the relation R^P is an equivalence relation

I Using some structural properties of F , study the space X = P/R^P or other objects associated to it (for example, its group of homeomorphisms)

I Conversely, given a compact metrisable space X , find if there exists F , with limit P, such that P is a prespace whose quotient is X

General goal

I Start with a non-empty, at most countable, projective Fra¨ıss´e family F of finite topological LR-structures

I Compute its limit P

I Determine if P is a prespace; that is, determine if the relation R^P is an equivalence relation

I Using some structural properties of F , study the space X = P/R^P or other objects associated to it (for example, its group of homeomorphisms)

I Conversely, given a compact metrisable space X , find if there exists F , with limit P, such that P is a prespace whose quotient is X

General goal

I Start with a non-empty, at most countable, projective Fra¨ıss´e family F of finite topological LR-structures

I Compute its limit P

I Determine if P is a prespace; that is, determine if the relation R^P is an equivalence relation

I Using some structural properties of F , study the space X = P/R^P or other objects associated to it (for example, its group of homeomorphisms)

I Conversely, given a compact metrisable space X , find if there exists F , with limit P, such that P is a prespace whose quotient is X

General goal

I Start with a non-empty, at most countable, projective Fra¨ıss´e family F of finite topological LR-structures

I Compute its limit P

I Determine if P is a prespace; that is, determine if the relation R^P is an equivalence relation

I Using some structural properties of F , study the space X = P/R^P or other objects associated to it (for example, its group of homeomorphisms)

I Conversely, given a compact metrisable space X , find if there exists F , with limit P, such that P is a prespace whose quotient is X

General goal

I Start with a non-empty, at most countable, projective Fra¨ıss´e family F of finite topological LR-structures

I Compute its limit P

I Determine if P is a prespace; that is, determine if the relation R^P is an equivalence relation

I Using some structural properties of F , study the space X = P/R^P

or other objects associated to it (for example, its group of homeomorphisms)

I Conversely, given a compact metrisable space X , find if there exists F , with limit P, such that P is a prespace whose quotient is X

General goal

I Start with a non-empty, at most countable, projective Fra¨ıss´e family F of finite topological LR-structures

I Compute its limit P

I Determine if P is a prespace; that is, determine if the relation R^P is an equivalence relation

I Using some structural properties of F , study the space X = P/R^P or other objects associated to it (for example, its group of homeomorphisms)

I Conversely, given a compact metrisable space X , find if there exists F , with limit P, such that P is a prespace whose quotient is X

General goal

I Start with a non-empty, at most countable, projective Fra¨ıss´e family F of finite topological LR-structures

I Compute its limit P

I Determine if P is a prespace; that is, determine if the relation R^P is an equivalence relation

I Using some structural properties of F , study the space X = P/R^P or other objects associated to it (for example, its group of homeomorphisms)

I Conversely, given a compact metrisable space X , find if there exists F , with limit P, such that P is a prespace whose quotient is X

Example: the pseudo-arc (Irwin, Solecki - 2006)

I LR = {R}

I F = {finite linear graphs}

Theorem

I F is a projective Fra¨ıss´e family

I If P is the projective Fra¨ıss´e limit of F, then P/R^P is a prespace and P/R^P is a pseudo-arc

Example: the pseudo-arc (Irwin, Solecki - 2006)

I LR = {R}

I F = {finite linear graphs}

Theorem

I F is a projective Fra¨ıss´e family

I If P is the projective Fra¨ıss´e limit of F, then P/R^P is a prespace and P/R^P is a pseudo-arc

Example: the pseudo-arc (Irwin, Solecki - 2006)

I LR = {R}

I F = {finite linear graphs}

Theorem

I F is a projective Fra¨ıss´e family

I If P is the projective Fra¨ıss´e limit of F, then P/R^P is a prespace and P/R^P is a pseudo-arc

Example: the pseudo-arc (Irwin, Solecki - 2006)

I LR = {R}

I F = {finite linear graphs}

Theorem

I F is a projective Fra¨ıss´e family

I If P is the projective Fra¨ıss´e limit of F, then P/R^P is a prespace

and P/R^P is a pseudo-arc

Example: the pseudo-arc (Irwin, Solecki - 2006)

I LR = {R}

I F = {finite linear graphs}

Theorem

I F is a projective Fra¨ıss´e family

I If P is the projective Fra¨ıss´e limit of F, then P/R^P is a prespace and P/R^P is a pseudo-arc

Quotients of projective Fra¨ıss´e limits for L_R = {R}

Theorem

The following is the list of all spaces of the form P/R^P, where P is a prespace that is the projective Fra¨ıss´e limit of a projective Fra¨ıss´e family of finite topological {R}-structures.

I Cantor space

I disjoint sums of m singletons and n pseudo-arcs, with m + n > 0

I disjoint sums of n spaces, each of the form X = P ∪S

j ∈NQj, where:

I P is a pseudo-arc

I every Qj is a Cantor space clopen in X

I S

j ∈NQj is dense in X

Quotients of projective Fra¨ıss´e limits for L_R = {R}

Theorem

The following is the list of all spaces of the form P/R^P, where P is a prespace that is the projective Fra¨ıss´e limit of a projective Fra¨ıss´e family of finite topological {R}-structures.

I Cantor space

I disjoint sums of m singletons and n pseudo-arcs, with m + n > 0

I disjoint sums of n spaces, each of the form X = P ∪S

j ∈NQj, where:

I P is a pseudo-arc

I every Qj is a Cantor space clopen in X

I S

j ∈NQj is dense in X

Quotients of projective Fra¨ıss´e limits for L_R = {R}

Theorem

The following is the list of all spaces of the form P/R^P, where P is a prespace that is the projective Fra¨ıss´e limit of a projective Fra¨ıss´e family of finite topological {R}-structures.

I Cantor space

I disjoint sums of m singletons and n pseudo-arcs, with m + n > 0

I disjoint sums of n spaces, each of the form X = P ∪S

j ∈NQj, where:

I P is a pseudo-arc

I every Qj is a Cantor space clopen in X

I S

j ∈NQj is dense in X

Quotients of projective Fra¨ıss´e limits for L_R = {R}

Theorem

The following is the list of all spaces of the form P/R^P, where P is a prespace that is the projective Fra¨ıss´e limit of a projective Fra¨ıss´e family of finite topological {R}-structures.

I Cantor space

I disjoint sums of m singletons and n pseudo-arcs, with m + n > 0

I disjoint sums of n spaces, each of the form X = P ∪S

j ∈NQj, where:

I P is a pseudo-arc

I every Qj is a Cantor space clopen in X

I S

j ∈NQj is dense in X

Quotients of projective Fra¨ıss´e limits for L_R = {R}

Theorem

The following is the list of all spaces of the form P/R^P, where P is a prespace that is the projective Fra¨ıss´e limit of a projective Fra¨ıss´e family of finite topological {R}-structures.

I Cantor space

I disjoint sums of m singletons and n pseudo-arcs, with m + n > 0

I disjoint sums of n spaces, each of the form X = P ∪S

j ∈NQj, where:

I P is a pseudo-arc

I every Qj is a Cantor space clopen in X

I S

j ∈NQj is dense in X

Quotients of projective Fra¨ıss´e limits for L_R = {R}

Theorem

The following is the list of all spaces of the form P/R^P, where P is a prespace that is the projective Fra¨ıss´e limit of a projective Fra¨ıss´e family of finite topological {R}-structures.

I Cantor space

I disjoint sums of m singletons and n pseudo-arcs, with m + n > 0

I disjoint sums of n spaces, each of the form X = P ∪S

j ∈NQj, where:

I P is a pseudo-arc

I every Qj is a Cantor space clopen in X

I S

j ∈NQj is dense in X

Quotients of projective Fra¨ıss´e limits for L_R = {R}

Theorem

The following is the list of all spaces of the form P/R^P, where P is a prespace that is the projective Fra¨ıss´e limit of a projective Fra¨ıss´e family of finite topological {R}-structures.

I Cantor space

I disjoint sums of m singletons and n pseudo-arcs, with m + n > 0

I disjoint sums of n spaces, each of the form X = P ∪S

j ∈NQj, where:

I P is a pseudo-arc

I every Qj is a Cantor space clopen in X

I S

j ∈NQj is dense in X