Philip Doi Some Notes 1. Introduction Definition (From [2, 4, 5]) - - PowerPoint PPT Presentation

A Toronto Space

Philip Doi

Some Notes

1.

2.

Introduction

Definition

(From [2, 4, 5]) A topological space X will be called Toronto if it is homeomorphic to each of its full cardinality subspaces, which is to say: for every subspace S ⊂ X, if |S| = |X| , then S ∼ = X

2.

Introduction

Definition

(From [2, 4, 5]) A topological space X will be called Toronto if it is homeomorphic to each of its full cardinality subspaces, which is to say: for every subspace S ⊂ X, if |S| = |X| , then S ∼ = X

Note

An infinite space X, with |X| ℵ1, is Toronto if and only if X → (top X)1

ℵ0.

3.

Introduction

...related to the Toronto seminar problem

• f

whether there is an uncountable non-discrete space which is homeomorphic to each of its uncountable sub-

• spaces. There are rules for working on this latter prob-

lem.

3.

Introduction

...related to the Toronto seminar problem

• f

whether there is an uncountable non-discrete space which is homeomorphic to each of its uncountable sub-

• spaces. There are rules for working on this latter prob-
• lem. The problem can be worked upon only in groups of

three or more mathematicians,

3.

Introduction

...related to the Toronto seminar problem

• f

whether there is an uncountable non-discrete space which is homeomorphic to each of its uncountable sub-

• spaces. There are rules for working on this latter prob-
• lem. The problem can be worked upon only in groups of

three or more mathematicians, and it is required that alcohol,

3.

Introduction

...related to the Toronto seminar problem

• f

whether there is an uncountable non-discrete space which is homeomorphic to each of its uncountable sub-

• spaces. There are rules for working on this latter prob-
• lem. The problem can be worked upon only in groups of

three or more mathematicians, and it is required that alcohol, preferably beer,

3.

Introduction

...related to the Toronto seminar problem

• f

whether there is an uncountable non-discrete space which is homeomorphic to each of its uncountable sub-

• spaces. There are rules for working on this latter prob-
• lem. The problem can be worked upon only in groups of

three or more mathematicians, and it is required that alcohol, preferably beer, be present during this time.

3.

Introduction

...related to the Toronto seminar problem

• f

whether there is an uncountable non-discrete space which is homeomorphic to each of its uncountable sub-

• spaces. There are rules for working on this latter prob-
• lem. The problem can be worked upon only in groups of

three or more mathematicians, and it is required that alcohol, preferably beer, be present during this time. Contact anyone in the Toronto Set Theory Seminar for the current status of the problem. It may never be solved.

4.

Introduction

Examples

There are many examples of Toronto spaces, albeit trivial ones.

4.

Introduction

Examples

There are many examples of Toronto spaces, albeit trivial ones. The discrete topology.

4.

Introduction

Examples

There are many examples of Toronto spaces, albeit trivial ones. The discrete topology. The cofinite topology.

4.

Introduction

Examples

There are many examples of Toronto spaces, albeit trivial ones. The discrete topology. The cofinite topology. The indiscrete topology.

4.

Introduction

Examples

There are many examples of Toronto spaces, albeit trivial ones. The discrete topology. The cofinite topology. The indiscrete topology. The upper and lower topology on N, where the basis for each topology consists of sets in the form [0, n] and [n, ∞), respectively.

4.

Introduction

Examples

There are many examples of Toronto spaces, albeit trivial ones. The discrete topology. The cofinite topology. The indiscrete topology. The upper and lower topology on N, where the basis for each topology consists of sets in the form [0, n] and [n, ∞), respectively.

Remark

Up to homeomorphism, these examples are the only Toronto spaces of cardinality ℵ0.

5.

The Cantor Rank and Scatter Spaces

Definition

For every topological space X, we can define, via transfinite recursion, an ordinal sequence of decreasing closed sets:

5.

The Cantor Rank and Scatter Spaces

Definition

For every topological space X, we can define, via transfinite recursion, an ordinal sequence of decreasing closed sets: X0 = X

5.

The Cantor Rank and Scatter Spaces

Definition

For every topological space X, we can define, via transfinite recursion, an ordinal sequence of decreasing closed sets: X0 = X Xα+1 = Xα IX

α , where IX α is the set of all isolated points from

Xα.

5.

The Cantor Rank and Scatter Spaces

Definition

For every topological space X, we can define, via transfinite recursion, an ordinal sequence of decreasing closed sets: X0 = X Xα+1 = Xα IX

α , where IX α is the set of all isolated points from

Xα. Finally, where δ is a limit ordinal (not a successor to any

• rdinal α ∈ δ), then

Xδ =

• α∈δ

Xα

6.

The Cantor Rank

Definitions

The rank of a space X is the least ordinal δ such that Xδ = Xδ+1. In this case we say δ = rk (X)

6.

The Cantor Rank

Definitions

The rank of a space X is the least ordinal δ such that Xδ = Xδ+1. In this case we say δ = rk (X) We define the width of a space like so: wd (X) = sup

δ∈∞

• IX

δ

7.

Scattered Spaces

Definition

A topological space is said to be scattered if all of its nonvoid subspaces have isolated points in their subspace topology.

7.

Scattered Spaces

Definition

A topological space is said to be scattered if all of its nonvoid subspaces have isolated points in their subspace topology.

Lemma

A space is scattered if and only if X =

• δ∈rk(X)

IX

δ

8.

Historical Note

The study of scattered spaces began with Georg Cantor investigation of sets of uniqueness ( ¨ Uber die Ausdehnung eines Staze aus der Theorie der trigometrischen Reihen).

8.

Historical Note

The study of scattered spaces began with Georg Cantor investigation of sets of uniqueness ( ¨ Uber die Ausdehnung eines Staze aus der Theorie der trigometrischen Reihen). This pertains to the convergence of various Fourier series:

∞

• n=−∞

eint

• T

f (z) e−inzdz =

∞

• n=−∞

ˆ f (n) eint

8.

Historical Note

The study of scattered spaces began with Georg Cantor investigation of sets of uniqueness ( ¨ Uber die Ausdehnung eines Staze aus der Theorie der trigometrischen Reihen). This pertains to the convergence of various Fourier series:

∞

• n=−∞

eint

• T

f (z) e−inzdz =

∞

• n=−∞

ˆ f (n) eint See [3].

9.

Hausdorff Toronto Space

Proposition

An infinite, Hausdorff, Toronto Space X has infinitely many isolated points

9.

Hausdorff Toronto Space

Proposition

An infinite, Hausdorff, Toronto Space X has infinitely many isolated points

Proof.

(sketch) If x, y ∈ X are distinct, then there are two neighborhoods U and V separating them.

9.

Hausdorff Toronto Space

Proposition

An infinite, Hausdorff, Toronto Space X has infinitely many isolated points

Proof.

(sketch) If x, y ∈ X are distinct, then there are two neighborhoods U and V separating them. The Pigeon Hole Principle says |U| = |X| or |X U| = |X| .

9.

Hausdorff Toronto Space

Proposition

An infinite, Hausdorff, Toronto Space X has infinitely many isolated points

Proof.

(sketch) If x, y ∈ X are distinct, then there are two neighborhoods U and V separating them. The Pigeon Hole Principle says |U| = |X| or |X U| = |X| . In the first case, y is isolated in U ∪ {y}, whilst in the second, x is isolated in (X U) ∪ {x}.

9.

Hausdorff Toronto Space

Proposition

An infinite, Hausdorff, Toronto Space X has infinitely many isolated points

Proof.

(sketch) If x, y ∈ X are distinct, then there are two neighborhoods U and V separating them. The Pigeon Hole Principle says |U| = |X| or |X U| = |X| . In the first case, y is isolated in U ∪ {y}, whilst in the second, x is isolated in (X U) ∪ {x}. Lastly, we note that the complement of a finite set of isolated points is open in X.

10.

Hausdorff Toronto Space

Corollary

If X is Hausdorff and Toronto with cardinality ℵ0, then X is discrete.

10.

Hausdorff Toronto Space

Corollary

If X is Hausdorff and Toronto with cardinality ℵ0, then X is discrete.

Proposition

If X is Hausdorff, Toronto Space, then IX

0 = X.

10.

Hausdorff Toronto Space

Corollary

If X is Hausdorff and Toronto with cardinality ℵ0, then X is discrete.

Proposition

If X is Hausdorff, Toronto Space, then IX

0 = X.

Note

We just need

• IX
• = |X|, using the fact that IIX

= IX

11.

Hausdorff Toronto Space

Proposition

If X is non-discrete Toronto space, with cardinality ℵ1, then X is scattered rk (X) = ω1 and wd (X) = ℵ0.

12.

Hausdorff Toronto Space

Proposition

If X is a non-discrete Hausdorff Toronto space of cardinality ℵα, where α > 0, then there exists an ordinal γ ∈ α such that 2ℵγ = 2ℵα.

13.

The True Problem

Problem

Does there consistently exist a non-discrete Hausdorff Toronto space of cardinality ℵα? Let ÓαX be the supposition that X is such a space. Ó

13.

The True Problem

Problem

Does there consistently exist a non-discrete Hausdorff Toronto space of cardinality ℵα? Let ÓαX be the supposition that X is such a space.

Remarks:

Con (ZFC) = ⇒ Con ((∀α ∈ ∞ : ∀X : ¬ÓαX) + ZFC) ,

13.

The True Problem

Problem

Does there consistently exist a non-discrete Hausdorff Toronto space of cardinality ℵα? Let ÓαX be the supposition that X is such a space.

Remarks:

Con (ZFC) = ⇒ Con ((∀α ∈ ∞ : ∀X : ¬ÓαX) + ZFC) , which can be shown by assuming the Generalized Continuum Hypothesis: That ℵα+1 = 2ℵα for all α ∈ ∞.

14.

Historical Notes on the Problem

14.

Historical Notes on the Problem

Posed in the 90’s by Juris Steprans.

14.

Historical Notes on the Problem

Posed in the 90’s by Juris Steprans. Definition of Toronto Space appears in Finite and Infinite Combinatorics in Sets and Logic.

14.

Historical Notes on the Problem

Posed in the 90’s by Juris Steprans. Definition of Toronto Space appears in Finite and Infinite Combinatorics in Sets and Logic. In 2012, Vienna, Frank Tall discusses Topological Problems for Set Theorists.

14.

Historical Notes on the Problem

Posed in the 90’s by Juris Steprans. Definition of Toronto Space appears in Finite and Infinite Combinatorics in Sets and Logic. In 2012, Vienna, Frank Tall discusses Topological Problems for Set Theorists. Part of a series of “Toronto Problems” in Open Problems in Topology.

14.

Historical Notes on the Problem

Posed in the 90’s by Juris Steprans. Definition of Toronto Space appears in Finite and Infinite Combinatorics in Sets and Logic. In 2012, Vienna, Frank Tall discusses Topological Problems for Set Theorists. Part of a series of “Toronto Problems” in Open Problems in Topology. February 2014, William Brian’s paper appears.

15.

Status of the Problem

As far as I know, there is nothing new to report – the problem is still wide open, and if anyone has made further progress then they haven’t told me. It is cer- tainly a fun problem to think about, so I’m glad you had the chance to explore it a bit.

16.

The Toronto Problem

Folk Lore (to my knowledge): Ó

16.

The Toronto Problem

Folk Lore (to my knowledge):

It is unknown if there are regular, Tychonoff, or normal Toronto spaces.

Ó

16.

The Toronto Problem

Folk Lore (to my knowledge):

It is unknown if there are regular, Tychonoff, or normal Toronto spaces. Every T2, nondiscrete, Toronto space is not metrizable, compact, paracompact, metacompact, or Lindel¨

• f.

Ó

16.

The Toronto Problem

Folk Lore (to my knowledge):

It is unknown if there are regular, Tychonoff, or normal Toronto spaces. Every T2, nondiscrete, Toronto space is not metrizable, compact, paracompact, metacompact, or Lindel¨

• f.

If X is Tychonoff, then its Stone-Cech compactification βX is not metrizable, and the associated C∗-algebras Cb (X) ∼ = C (βX) are not separable.

Ó

16.

The Toronto Problem

Folk Lore (to my knowledge):

It is unknown if there are regular, Tychonoff, or normal Toronto spaces. Every T2, nondiscrete, Toronto space is not metrizable, compact, paracompact, metacompact, or Lindel¨

• f.

If X is Tychonoff, then its Stone-Cech compactification βX is not metrizable, and the associated C∗-algebras Cb (X) ∼ = C (βX) are not separable.

From Kenneth Kunen, ω + 1 cannot be embedded into X, where Ó1X.

17.

On the function space

From [1].

Ó

17.

On the function space

From [1].

Proposition

Suppose X is a Tychonoff space. Then the C∗-algebra C (βX) is separable if and only if X is compact and metrizable.

Ó

17.

On the function space

From [1].

Proposition

Suppose X is a Tychonoff space. Then the C∗-algebra C (βX) is separable if and only if X is compact and metrizable.

The study of X, where Ó1X, has been the main focus, since the set of isolated points is dense. The homeomorphisms on IX

0 are determined by their action on

IX

0 . Furthermore, a compactification of X is a

compactification of IX

0 .

18.

Some Observations About βX

Suppose Ó1X, and assume X is Tychonoff. Then the following holds

18.

Some Observations About βX

Suppose Ó1X, and assume X is Tychonoff. Then the following holds: IX

0 = IβX

18.

Some Observations About βX

Suppose Ó1X, and assume X is Tychonoff. Then the following holds: IX

0 = IβX

Since IX

0 is dense in βX, βX is a quotient of βIX 0 .

18.

Some Observations About βX

Suppose Ó1X, and assume X is Tychonoff. Then the following holds: IX

0 = IβX

Since IX

0 is dense in βX, βX is a quotient of βIX 0 .

βIX

0 is βN (a subject of research in set theory).

18.

Some Observations About βX

Suppose Ó1X, and assume X is Tychonoff. Then the following holds: IX

0 = IβX

Since IX

0 is dense in βX, βX is a quotient of βIX 0 .

βIX

0 is βN (a subject of research in set theory).

If f : X − → X is an embedding such that f

• IX
• = IX

0 ,

then fβ is onto.

18.

Some Observations About βX

Suppose Ó1X, and assume X is Tychonoff. Then the following holds: IX

0 = IβX

Since IX

0 is dense in βX, βX is a quotient of βIX 0 .

βIX

0 is βN (a subject of research in set theory).

If f : X − → X is an embedding such that f

• IX
• = IX

0 ,

then fβ is onto. Related to dynamics on βN.

18.

Some Observations About βX

Suppose Ó1X, and assume X is Tychonoff. Then the following holds: IX

0 = IβX

Since IX

0 is dense in βX, βX is a quotient of βIX 0 .

βIX

0 is βN (a subject of research in set theory).

If f : X − → X is an embedding such that f

• IX
• = IX

0 ,

then fβ is onto. Related to dynamics on βN. Note that the dynamics of a semigroup S can be extended to βS.

18.

Some Observations About βX

Suppose Ó1X, and assume X is Tychonoff. Then the following holds: IX

0 = IβX

Since IX

0 is dense in βX, βX is a quotient of βIX 0 .

βIX

0 is βN (a subject of research in set theory).

If f : X − → X is an embedding such that f

• IX
• = IX

0 ,

then fβ is onto. Related to dynamics on βN. Note that the dynamics of a semigroup S can be extended to βS.

19.

Further observations

Suppose Ó1X. Consider the family F of f : X − → X such that f (X) = X

19.

Further observations

Suppose Ó1X. Consider the family F of f : X − → X such that f (X) = X

IX

0 ⊂ f (X) (meaning IX 0 = If(X)

)

19.

Further observations

Suppose Ó1X. Consider the family F of f : X − → X such that f (X) = X

IX

0 ⊂ f (X) (meaning IX 0 = If(X)

)

x → f (x) : X − → f (X) is a homeomorphism.

19.

Further observations

Suppose Ó1X. Consider the family F of f : X − → X such that f (X) = X

IX

0 ⊂ f (X) (meaning IX 0 = If(X)

)

x → f (x) : X − → f (X) is a homeomorphism.

Note that f ↾ IX

0 ∈ Aut

• IX
• .

19.

Further observations

Suppose Ó1X. Consider the family F of f : X − → X such that f (X) = X

IX

0 ⊂ f (X) (meaning IX 0 = If(X)

)

x → f (x) : X − → f (X) is a homeomorphism.

Note that f ↾ IX

0 ∈ Aut

• IX
• .

20.

A Semigroup

Where S =

• f ↾ IX
• f ∈ F
• , S is a left-cancellable,

subsemigroup of Aut

• IX
• . Furthermore, S is an ideal for the

set

• f ↾ IX
• f : X −

→ X is an embedding, f

• IX
• = IX
• .

20.

A Semigroup

Where S =

• f ↾ IX
• f ∈ F
• , S is a left-cancellable,

subsemigroup of Aut

• IX
• . Furthermore, S is an ideal for the

set

• f ↾ IX
• f : X −

→ X is an embedding, f

• IX
• = IX
• .

The follow is true:

20.

A Semigroup

Where S =

• f ↾ IX
• f ∈ F
• , S is a left-cancellable,

subsemigroup of Aut

• IX
• . Furthermore, S is an ideal for the

set

• f ↾ IX
• f : X −

→ X is an embedding, f

• IX
• = IX
• .

The follow is true: S is a pure semigroup.

20.

A Semigroup

Where S =

• f ↾ IX
• f ∈ F
• , S is a left-cancellable,

subsemigroup of Aut

• IX
• . Furthermore, S is an ideal for the

set

• f ↾ IX
• f : X −

→ X is an embedding, f

• IX
• = IX
• .

The follow is true: S is a pure semigroup. There are no inverses.

20.

A Semigroup

Where S =

• f ↾ IX
• f ∈ F
• , S is a left-cancellable,

subsemigroup of Aut

• IX
• . Furthermore, S is an ideal for the

set

• f ↾ IX
• f : X −

→ X is an embedding, f

• IX
• = IX
• .

The follow is true: S is a pure semigroup. There are no inverses. |S| 2ℵ0 and |S| = 2ℵ1.

20.

A Semigroup

Where S =

• f ↾ IX
• f ∈ F
• , S is a left-cancellable,

subsemigroup of Aut

• IX
• . Furthermore, S is an ideal for the

set

• f ↾ IX
• f : X −

→ X is an embedding, f

• IX
• = IX
• .

The follow is true: S is a pure semigroup. There are no inverses. |S| 2ℵ0 and |S| = 2ℵ1.

21.

Some subsemigroups

Where Y ∈ [X]ℵ1 such that IX

0 ⊂ Y , we can define

SY =

• f ↾ IX
• f ∈ F, f (X) ⊂ Y
• .

21.

Some subsemigroups

Where Y ∈ [X]ℵ1 such that IX

0 ⊂ Y , we can define

SY =

• f ↾ IX
• f ∈ F, f (X) ⊂ Y
• .

and for every T ⊂ S, SY T ⊂ SY

21.

Some subsemigroups

Where Y ∈ [X]ℵ1 such that IX

0 ⊂ Y , we can define

SY =

• f ↾ IX
• f ∈ F, f (X) ⊂ Y
• .

and for every T ⊂ S, SY T ⊂ SY If A ⊂ B, then SA ⊂ SB.

21.

Some subsemigroups

Where Y ∈ [X]ℵ1 such that IX

0 ⊂ Y , we can define

SY =

• f ↾ IX
• f ∈ F, f (X) ⊂ Y
• .

and for every T ⊂ S, SY T ⊂ SY If A ⊂ B, then SA ⊂ SB. Also, where A =

• i∈J

Ai,

21.

Some subsemigroups

Where Y ∈ [X]ℵ1 such that IX

0 ⊂ Y , we can define

SY =

• f ↾ IX
• f ∈ F, f (X) ⊂ Y
• .

and for every T ⊂ S, SY T ⊂ SY If A ⊂ B, then SA ⊂ SB. Also, where A =

• i∈J

Ai, we have SA =

• i∈J

Si.

22.

Representations

These semigroups can be represented as isometries on ℓ2 IX

• .

22.

Representations

These semigroups can be represented as isometries on ℓ2 IX

• .

We have can induce a linear isometry through δx → δσ(x) where σ ∈ S

22.

Representations

These semigroups can be represented as isometries on ℓ2 IX

• .

We have can induce a linear isometry through δx → δσ(x) where σ ∈ S

The study of representations of semigroups is somewhat restricted,

22.

Representations

These semigroups can be represented as isometries on ℓ2 IX

• .

We have can induce a linear isometry through δx → δσ(x) where σ ∈ S

The study of representations of semigroups is somewhat restricted, focusing on nicer cases such as single parameter semigroups.

23.

Groupoids

We can construct internal topological groupoids from the subspaces of X. While some of these can be restricted to permutations on IX

0 , the mapping is not guaranteed to be

injective.

24.

More constrained functions

Where LX

α =

• γ∈α+1

IX

γ ,

we can define Sα =

• f ↾ IX
• f ∈ F, LX

α ⊂ f (X)

• .

24.

More constrained functions

Where LX

α =

• γ∈α+1

IX

γ ,

we can define Sα =

• f ↾ IX
• f ∈ F, LX

α ⊂ f (X)

• .

The spaces LX

α are countable scatter spaces of rank of

α.

24.

More constrained functions

Where LX

α =

• γ∈α+1

IX

γ ,

we can define Sα =

• f ↾ IX
• f ∈ F, LX

α ⊂ f (X)

• .

The spaces LX

α are countable scatter spaces of rank of

α.Through induction, LX

α = Lf(X) α

, where f ↾ IX

0 ∈ Sα.

25.

Questions?

26.

References

• J. Bell.

The stone-cech compactification of tychonoff spaces. 2014. William Rea Brian. The toronto problem. Topology and its Applications. Thomas W. K¨

• rner.

Sets of uniqueness. Cahiers du s´ eminaire d’histoire des math´ ematiques, Ser. 2, 2:51–63, 1992. Elliott Pearl. Open problems in topology. Topology and its Applications, 136(1):37–85, 2004. Elliott M Pearl.