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

A Toronto Space Philip Doi Some Notes 1.

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 | � ℵ 1 , is Toronto if and only if X → (top X ) 1 ℵ 0 . 2.

Introduction ...related to the Toronto seminar problem of 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 of 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 of 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 of 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 of 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 of 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. 3.

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 ℵ 0 . 4.

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: X 0 = 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: X 0 = X X α +1 = X α � I X α , where I X α is the set of all isolated points from X α . 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: X 0 = X X α +1 = X α � I X α , where I X α is the set of all isolated points from X α . Finally, where δ is a limit ordinal (not a successor to any ordinal α ∈ δ ), then X δ = � X α α ∈ δ 5.

The Cantor Rank Definitions The rank of a space X is the least ordinal δ such that X δ = X δ +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 X δ = X δ +1 . In this case we say δ = rk ( X ) We define the width of a space like so: � � I X � wd ( X ) = sup δ � δ ∈∞ 6.

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 � I X X = δ δ ∈ rk( X ) 7.

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: ∞ ∞ � ˆ � e int f ( z ) e − inz dz = � f ( n ) e int T n = −∞ n = −∞ 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: ∞ ∞ � ˆ � e int f ( z ) e − inz dz = � f ( n ) e int T n = −∞ n = −∞ See [3]. 8.

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 . 9.

Hausdorff Toronto Space Corollary If X is Hausdorff and Toronto with cardinality ℵ 0 , then X is discrete. 10.

Hausdorff Toronto Space Corollary If X is Hausdorff and Toronto with cardinality ℵ 0 , then X is discrete. Proposition If X is Hausdorff, Toronto Space, then I X 0 = X. 10.

Hausdorff Toronto Space Corollary If X is Hausdorff and Toronto with cardinality ℵ 0 , then X is discrete. Proposition If X is Hausdorff, Toronto Space, then I X 0 = X. Note � � � = | X | , using the fact that I I X � I X = I X We just need 0 � � 0 0 0 10.

Hausdorff Toronto Space Proposition If X is non-discrete Toronto space, with cardinality ℵ 1 , then X is scattered rk ( X ) = ω 1 and wd ( X ) = ℵ 0 . 11.

Hausdorff Toronto Space Proposition If X is a non-discrete Hausdorff Toronto space of cardinality ℵ α , where α > 0 , then there exists an ordinal γ ∈ α such that 2 ℵ γ = 2 ℵ α . 12.

Ó The True Problem Problem Does there consistently exist a non-discrete Hausdorff Toronto space of cardinality ℵ α ? Let Ó α X be the supposition that X is such a space. 13.