Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

There is no space-filling curve for a compact
connected nowhere separable linearlyordered topological space
Tetsuya Ishiu
Department of Mathematics
Miami University

November 17th, 2020

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

Outline
1

2

3
4

5

Introduction
What are the reals?
Nowhere separable LOTS
Coordinate-wise theorem
Statement
Ideas
Isn’t it nice?
Space-filling curves
Space-filling curves
Linearly Ordered Semigroups
Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups
Open Problems
Open Problems
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

The characterization of R
The real line, R, is the unique complete self-dense separable
linearly ordered sets without endpoints.
By the way, when I say the reals in this talk, I do not mean 2ω orω ω , but the genuine real line, which is complete.
A linearly ordered set can be topologized by the order topology.
Such a topological space is called a linearly ordered topologicalspace (LOTS).
The standard topology on R coincide with the order topology.
So, R is a LOTS.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

The characterization of R
The real line, R, is the unique complete self-dense separable
linearly ordered sets without endpoints.
By the way, when I say the reals in this talk, I do not mean 2ω orω ω , but the genuine real line, which is complete.
A linearly ordered set can be topologized by the order topology.
Such a topological space is called a linearly ordered topologicalspace (LOTS).
The standard topology on R coincide with the order topology.
So, R is a LOTS.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

The characterization of R
The real line, R, is the unique complete self-dense separable
linearly ordered sets without endpoints.
By the way, when I say the reals in this talk, I do not mean 2ω orω ω , but the genuine real line, which is complete.
A linearly ordered set can be topologized by the order topology.
Such a topological space is called a linearly ordered topologicalspace (LOTS).
The standard topology on R coincide with the order topology.
So, R is a LOTS.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

The characterization of R
The real line, R, is the unique complete self-dense separable
linearly ordered sets without endpoints.
By the way, when I say the reals in this talk, I do not mean 2ω orω ω , but the genuine real line, which is complete.
A linearly ordered set can be topologized by the order topology.
Such a topological space is called a linearly ordered topologicalspace (LOTS).
The standard topology on R coincide with the order topology.
So, R is a LOTS.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

Connectedness
Lemma
Let L be a LOTS. Then the following are equivalent.
1

L is connected as a topological space (i.e. there are no two
open sets U and W such that U ∩ W = ∅ and U ∪ W = L).

2

L is complete and self-dense.

So, R is characterized as aconnected,
separable,
linearly ordered space
without endpoints.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

Connectedness
Lemma
Let L be a LOTS. Then the following are equivalent.
1

L is connected as a topological space (i.e. there are no two
open sets U and W such that U ∩ W = ∅ and U ∪ W = L).

2

L is complete and self-dense.

So, R is characterized as aconnected,
separable,
linearly ordered space
without endpoints.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

Separability
R is characterized as aconnected,
separable,
linearly ordered space
without endpoints.
‘Without endpoints’ is a superficial condition.
‘Connectedness’ seems important to state continuous
phenomenon (I know there is an objection, but hold on).
Being ‘linearly ordered’ seems necessary to compare thequantities.
How about ‘separability’? What is wrong to abandon it?
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

Separability
R is characterized as aconnected,
separable,
linearly ordered space
without endpoints.
‘Without endpoints’ is a superficial condition.
‘Connectedness’ seems important to state continuous
phenomenon (I know there is an objection, but hold on).
Being ‘linearly ordered’ seems necessary to compare thequantities.
How about ‘separability’? What is wrong to abandon it?
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

Separability
R is characterized as aconnected,
separable,
linearly ordered space
without endpoints.
‘Without endpoints’ is a superficial condition.
‘Connectedness’ seems important to state continuous
phenomenon (I know there is an objection, but hold on).
Being ‘linearly ordered’ seems necessary to compare thequantities.
How about ‘separability’? What is wrong to abandon it?
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

Separability
R is characterized as aconnected,
separable,
linearly ordered space
without endpoints.
‘Without endpoints’ is a superficial condition.
‘Connectedness’ seems important to state continuous
phenomenon (I know there is an objection, but hold on).
Being ‘linearly ordered’ seems necessary to compare thequantities.
How about ‘separability’? What is wrong to abandon it?
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

Separability
R is characterized as aconnected,
separable,
linearly ordered space
without endpoints.
‘Without endpoints’ is a superficial condition.
‘Connectedness’ seems important to state continuous
phenomenon (I know there is an objection, but hold on).
Being ‘linearly ordered’ seems necessary to compare thequantities.
How about ‘separability’? What is wrong to abandon it?
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

Outline
1

2

3
4

5

Introduction
What are the reals?
Nowhere separable LOTS
Coordinate-wise theorem
Statement
Ideas
Isn’t it nice?
Space-filling curves
Space-filling curves
Linearly Ordered Semigroups
Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups
Open Problems
Open Problems
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

Nowhere separable spaces
Definition
A topological space X is nowhere separable if and only if nononempty open set is separable.
Then, a connected LOTS is nowhere separable if and only ifthere is no nonempty open interval that is homeomorphic to R.
Question
How similar to R can a connected nowhere separable LOTS
be?
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

Nowhere separable spaces
Definition
A topological space X is nowhere separable if and only if nononempty open set is separable.
Then, a connected LOTS is nowhere separable if and only ifthere is no nonempty open interval that is homeomorphic to R.
Question
How similar to R can a connected nowhere separable LOTS
be?
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

Nowhere separable spaces
Definition
A topological space X is nowhere separable if and only if nononempty open set is separable.
Then, a connected LOTS is nowhere separable if and only ifthere is no nonempty open interval that is homeomorphic to R.
Question
How similar to R can a connected nowhere separable LOTS
be?
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

ExamplesExampleLet κ be an uncountable regular cardinal. Define
L = { f ∈ 2κ | f is not eventually 1 }ordered by lexicographical ordering.
Then, L is a connected nowhere separable LOTS.
This set has a dense set of points of cofinality κ.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

ExamplesExampleLet κ be an uncountable regular cardinal. Define
L = { f ∈ 2κ | f is not eventually 1 }ordered by lexicographical ordering.
Then, L is a connected nowhere separable LOTS.
This set has a dense set of points of cofinality κ.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

ExamplesExampleLet κ be an uncountable regular cardinal. Define
L = { f ∈ 2κ | f is not eventually 1 }ordered by lexicographical ordering.
Then, L is a connected nowhere separable LOTS.
This set has a dense set of points of cofinality κ.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

Example (cont.)

ExampleLet L be a self-dense Aronszajn line and L̂ the Dedekind
completion of L.
Then, L̂ is a connected nowhere separable LOTS.
Such a linearly ordered set is called an Aronszajn continuum inTodorcevic’s article in the Handbook of Set Theoretic Topology,
but there seems not much research afterwards (correct me if I
am wrong).

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

Example (cont.)

ExampleLet L be a self-dense Aronszajn line and L̂ the Dedekind
completion of L.
Then, L̂ is a connected nowhere separable LOTS.
Such a linearly ordered set is called an Aronszajn continuum inTodorcevic’s article in the Handbook of Set Theoretic Topology,
but there seems not much research afterwards (correct me if I
am wrong).

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

What are the reals?
Nowhere separable LOTS

Example (cont.)

ExampleLet L be a self-dense Aronszajn line and L̂ the Dedekind
completion of L.
Then, L̂ is a connected nowhere separable LOTS.
Such a linearly ordered set is called an Aronszajn continuum inTodorcevic’s article in the Handbook of Set Theoretic Topology,
but there seems not much research afterwards (correct me if I
am wrong).

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

Outline
1

2

3
4

5

Introduction
What are the reals?
Nowhere separable LOTS
Coordinate-wise theorem
Statement
Ideas
Isn’t it nice?
Space-filling curves
Space-filling curves
Linearly Ordered Semigroups
Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups
Open Problems
Open Problems
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

The differences between R and connected nowhereseparable LOTS.
In fact, there are qualitative differences between R andconnected nowhere separable LOTS.
We shall mention two such differences.
Every continuous injection from the product into the
product is coordinate-wise.
There is no space-filling curve.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

The differences between R and connected nowhereseparable LOTS.
In fact, there are qualitative differences between R andconnected nowhere separable LOTS.
We shall mention two such differences.
Every continuous injection from the product into the
product is coordinate-wise.
There is no space-filling curve.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

Coordinate-wise Theorem
Theorem
Let n be a non-zero natural number, K0 , . . . , Kn−1 , L0 , . . . , Ln−1
be Q
connected Q
nowhere separable LOTS, andf : i<n Ki → i<n Li a continuous injection. Then, f iscoordinate-wise, namely there exist a bijection h : n → n and afunction
Q pi : Ki → Lh(i) for each i < n such that for allx ∈ i<n Ki and i < n, f (x)(h(i)) = pi (x(i))
This theorem extends the result of Eda and Kamijo about LOTS
that has densely many points of uncountable cofinality or
coinitiality.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

Coordinate-wise Theorem
Theorem
Let n be a non-zero natural number, K0 , . . . , Kn−1 , L0 , . . . , Ln−1
be Q
connected Q
nowhere separable LOTS, andf : i<n Ki → i<n Li a continuous injection. Then, f iscoordinate-wise, namely there exist a bijection h : n → n and afunction
Q pi : Ki → Lh(i) for each i < n such that for allx ∈ i<n Ki and i < n, f (x)(h(i)) = pi (x(i))
This theorem extends the result of Eda and Kamijo about LOTS
that has densely many points of uncountable cofinality or
coinitiality.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

Two-dimensional version
The previous theorem is not that intuitive, so let me state thetwo-dimensional version.
Theorem
Let K0 , K1 , L0 , L1 be connected nowhere separable LOTS. Thenevery continuous injection f : K0 × K1 → L0 × L1 iscoordinate-wise, namely either
1

there exist functions g0 : K0 → L0 and g1 : K1 → L1 suchthat f (x, y ) = hg0 (x), g1 (y )i, or2there exist functions g0 : K0 → L1 and g1 : K1 → L0 suchthat f (x, y ) = hg1 (y ), g0 (x)i.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

Outline
1

2

3
4

5

Introduction
What are the reals?
Nowhere separable LOTS
Coordinate-wise theorem
Statement
Ideas
Isn’t it nice?
Space-filling curves
Space-filling curves
Linearly Ordered Semigroups
Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups
Open Problems
Open Problems
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

Countable elementary submodels
Suppose that N is a countable elementary submodel of H(θ) forsome sufficiently large regular cardinal θ) such that everything
in the context belongs to N.
Since L is nowhere separable, for every nonempty open subset
U of L,U \ Cl(L ∩ N) 6= ∅This is a huge difference between separable and nowhereseparable topological spaces.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

Countable elementary submodels
Suppose that N is a countable elementary submodel of H(θ) forsome sufficiently large regular cardinal θ) such that everything
in the context belongs to N.
Since L is nowhere separable, for every nonempty open subset
U of L,U \ Cl(L ∩ N) 6= ∅This is a huge difference between separable and nowhereseparable topological spaces.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

Countable elementary submodels
Suppose that N is a countable elementary submodel of H(θ) forsome sufficiently large regular cardinal θ) such that everything
in the context belongs to N.
Since L is nowhere separable, for every nonempty open subset
U of L,U \ Cl(L ∩ N) 6= ∅This is a huge difference between separable and nowhereseparable topological spaces.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

Some notations
Let
J(L, N) = { x ∈ L \ Cl(L ∩ N) | ∃x0 , x1 ∈ L ∩ N(x0 < x < x1 ) }For each x̂ ∈ J(L, N), defineη(x̂) = η(L, N, x̂) = sup { x ∈ L ∩ N | x < x̂ }ζ(x̂) = ζ(L, N, x̂) = inf { x ∈ L ∩ N | x > x̂ }I(x̂) = I(L, N, x̂) = [η(L, N, x̂), ζ(L, N, x̂)]i.e. I(x̂) is the largest interval which has x̂ as an element andwhose interior is disjoint from N.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

Some notations
Let
J(L, N) = { x ∈ L \ Cl(L ∩ N) | ∃x0 , x1 ∈ L ∩ N(x0 < x < x1 ) }For each x̂ ∈ J(L, N), defineη(x̂) = η(L, N, x̂) = sup { x ∈ L ∩ N | x < x̂ }ζ(x̂) = ζ(L, N, x̂) = inf { x ∈ L ∩ N | x > x̂ }I(x̂) = I(L, N, x̂) = [η(L, N, x̂), ζ(L, N, x̂)]i.e. I(x̂) is the largest interval which has x̂ as an element andwhose interior is disjoint from N.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

Properties

Let g : K → L be a continuous function (with g ∈ N) andx̂ ∈ J(K , N).
Then, g has very nice properties on I(x̂).
g  I(x̂) has a maximum and a minimum at the endpoints.
If g(x̂) ∈ N, then g  I(x̂) is constant.
If g  I(x̂) is not a constant, then g(x̂) ∈ J(L, N) andg → I(x̂) = I(g(x̂)).

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

Properties

Let g : K → L be a continuous function (with g ∈ N) andx̂ ∈ J(K , N).
Then, g has very nice properties on I(x̂).
g  I(x̂) has a maximum and a minimum at the endpoints.
If g(x̂) ∈ N, then g  I(x̂) is constant.
If g  I(x̂) is not a constant, then g(x̂) ∈ J(L, N) andg → I(x̂) = I(g(x̂)).

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

Cell-to-cell
By using the facts in the previous slide, we can prove:Lemma
Let x̂0 ∈ J(K0 , N) and x̂1 ∈ J(K1 , N). Define hŷ0 , ŷ1 i = f (x̂0 , x̂1 ).
Then,
f → (I(x̂0 ) × I(x̂1 )) = I(ŷ0 ) × I(ŷ1 )(To be honest, it requires many technical arguments to get here
from the previous slide...)By using the previous lemma, we can prove that f iscoordinate-wise.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

Cell-to-cell
By using the facts in the previous slide, we can prove:Lemma
Let x̂0 ∈ J(K0 , N) and x̂1 ∈ J(K1 , N). Define hŷ0 , ŷ1 i = f (x̂0 , x̂1 ).
Then,
f → (I(x̂0 ) × I(x̂1 )) = I(ŷ0 ) × I(ŷ1 )(To be honest, it requires many technical arguments to get here
from the previous slide...)By using the previous lemma, we can prove that f iscoordinate-wise.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

Cell-to-cell
By using the facts in the previous slide, we can prove:Lemma
Let x̂0 ∈ J(K0 , N) and x̂1 ∈ J(K1 , N). Define hŷ0 , ŷ1 i = f (x̂0 , x̂1 ).
Then,
f → (I(x̂0 ) × I(x̂1 )) = I(ŷ0 ) × I(ŷ1 )(To be honest, it requires many technical arguments to get here
from the previous slide...)By using the previous lemma, we can prove that f iscoordinate-wise.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

Outline
1

2

3
4

5

Introduction
What are the reals?
Nowhere separable LOTS
Coordinate-wise theorem
Statement
Ideas
Isn’t it nice?
Space-filling curves
Space-filling curves
Linearly Ordered Semigroups
Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups
Open Problems
Open Problems
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

Power of countable elementary submodels

I really like the outline of this proof (as much as I hate annoying
details in it).
This has nothing to do with consistency, inner models, large
cardinals, and other ordinary set theory things, but it
demonstrates how strong elementary submodel arguments are.

Can’t we prove any other theorems this way?

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

Power of countable elementary submodels

I really like the outline of this proof (as much as I hate annoying
details in it).
This has nothing to do with consistency, inner models, large
cardinals, and other ordinary set theory things, but it
demonstrates how strong elementary submodel arguments are.

Can’t we prove any other theorems this way?

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

Power of countable elementary submodels

I really like the outline of this proof (as much as I hate annoying
details in it).
This has nothing to do with consistency, inner models, large
cardinals, and other ordinary set theory things, but it
demonstrates how strong elementary submodel arguments are.

Can’t we prove any other theorems this way?

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Statement
Ideas
Isn’t it nice?

Power of countable elementary submodels

I really like the outline of this proof (as much as I hate annoying
details in it).
This has nothing to do with consistency, inner models, large
cardinals, and other ordinary set theory things, but it
demonstrates how strong elementary submodel arguments are.

Can’t we prove any other theorems this way?

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

Outline
1

2

3
4

5

Introduction
What are the reals?
Nowhere separable LOTS
Coordinate-wise theorem
Statement
Ideas
Isn’t it nice?
Space-filling curves
Space-filling curves
Linearly Ordered Semigroups
Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups
Open Problems
Open Problems
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

Space-filling curves on R2

It is well-known that there exists a continuous surjection from[0, 1] onto [0, 1] × [0, 1]. Such a curve is called a space-filling
curve.
Because G. Peano discovered the first example, space-filling
curves are sometimes called Peano curves (at least according
to Wikipedia). But the Peano curve more often refers to theparticular example that G. Peano discovered.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

Space-filling curves on R2

It is well-known that there exists a continuous surjection from[0, 1] onto [0, 1] × [0, 1]. Such a curve is called a space-filling
curve.
Because G. Peano discovered the first example, space-filling
curves are sometimes called Peano curves (at least according
to Wikipedia). But the Peano curve more often refers to theparticular example that G. Peano discovered.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

What about connected nowhere separable LOTS?
It is natural to ask if there is a space-filling curve onto the
product of connected nowhere separable LOTS?
Question
Let K be a connected nowhere separable LOTS. Is there acontinuous surjection from K onto K × K ?
More generally: Let K , L0 , L1 be connected nowhere separable
LOTS. Is there a continuous surjection from K onto L0 × L1 ?

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

What about connected nowhere separable LOTS?
It is natural to ask if there is a space-filling curve onto the
product of connected nowhere separable LOTS?
Question
Let K be a connected nowhere separable LOTS. Is there acontinuous surjection from K onto K × K ?
More generally: Let K , L0 , L1 be connected nowhere separable
LOTS. Is there a continuous surjection from K onto L0 × L1 ?

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

What about connected nowhere separable LOTS?
It is natural to ask if there is a space-filling curve onto the
product of connected nowhere separable LOTS?
Question
Let K be a connected nowhere separable LOTS. Is there acontinuous surjection from K onto K × K ?
More generally: Let K , L0 , L1 be connected nowhere separable
LOTS. Is there a continuous surjection from K onto L0 × L1 ?

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves[0, 1] onto L0 × L1 ?
By the way, we can show that there is no space-filling curve
onto L0 × L1 in the original sense for connected nowhereseparable LOTS L0 , L1 . Namely,
Proposition
Let L0 , L1 be connected nowhere separable LOTS. Then, there
is no continuous surjection from [0, 1] onto L0 × L1 .
This is an easy corollary of the fact that every continuousfunction from an interval of R into a nowhere separable LOTS isconstant.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves[0, 1] onto L0 × L1 ?
By the way, we can show that there is no space-filling curve
onto L0 × L1 in the original sense for connected nowhereseparable LOTS L0 , L1 . Namely,
Proposition
Let L0 , L1 be connected nowhere separable LOTS. Then, there
is no continuous surjection from [0, 1] onto L0 × L1 .
This is an easy corollary of the fact that every continuousfunction from an interval of R into a nowhere separable LOTS isconstant.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves[0, 1] onto L0 × L1 ?
By the way, we can show that there is no space-filling curve
onto L0 × L1 in the original sense for connected nowhereseparable LOTS L0 , L1 . Namely,
Proposition
Let L0 , L1 be connected nowhere separable LOTS. Then, there
is no continuous surjection from [0, 1] onto L0 × L1 .
This is an easy corollary of the fact that every continuousfunction from an interval of R into a nowhere separable LOTS isconstant.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

K onto L0 × L1 ?
So, it is more reasonable to ask if there is a continuous
surjection from K onto L0 × L1 .
As the title of this talk suggests, the answer is NO if K iscompact.
Theorem
Let K , L0 , L1 be connected nowhere separable LOTS such that
K is compact. Then, there exists no continuous surjection fromK onto L0 × L1 .
This can be proved by the lemmas for the ‘Coordinate-wise’
theorem.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

K onto L0 × L1 ?
So, it is more reasonable to ask if there is a continuous
surjection from K onto L0 × L1 .
As the title of this talk suggests, the answer is NO if K iscompact.
Theorem
Let K , L0 , L1 be connected nowhere separable LOTS such that
K is compact. Then, there exists no continuous surjection fromK onto L0 × L1 .
This can be proved by the lemmas for the ‘Coordinate-wise’
theorem.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

K onto L0 × L1 ?
So, it is more reasonable to ask if there is a continuous
surjection from K onto L0 × L1 .
As the title of this talk suggests, the answer is NO if K iscompact.
Theorem
Let K , L0 , L1 be connected nowhere separable LOTS such that
K is compact. Then, there exists no continuous surjection fromK onto L0 × L1 .
This can be proved by the lemmas for the ‘Coordinate-wise’
theorem.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

K onto L0 × L1 ?
So, it is more reasonable to ask if there is a continuous
surjection from K onto L0 × L1 .
As the title of this talk suggests, the answer is NO if K iscompact.
Theorem
Let K , L0 , L1 be connected nowhere separable LOTS such that
K is compact. Then, there exists no continuous surjection fromK onto L0 × L1 .
This can be proved by the lemmas for the ‘Coordinate-wise’
theorem.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

Proof
Proof.
Suppose that there exists a continuous surjection
f : K → L0 × L1 . Let g0 , g1 be its component functions, i.e.
f (t) = hg0 (t), g1 (t)i for all t ∈ K .
Let N be a countable elementary submodel of H(θ) for somesufficiently large regular cardinal θ with K , L0 , L1 , f ∈ N
Let hxi |i < ωi be a sequence of distinct elements of L0 ∩ N andy ∈ J(L1 , N). Since f is surjective, for each i < ω, there existti ∈ K such that f (ti ) = hxi , y i.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

Proof
Proof.
Suppose that there exists a continuous surjection
f : K → L0 × L1 . Let g0 , g1 be its component functions, i.e.
f (t) = hg0 (t), g1 (t)i for all t ∈ K .
Let N be a countable elementary submodel of H(θ) for somesufficiently large regular cardinal θ with K , L0 , L1 , f ∈ N
Let hxi |i < ωi be a sequence of distinct elements of L0 ∩ N andy ∈ J(L1 , N). Since f is surjective, for each i < ω, there existti ∈ K such that f (ti ) = hxi , y i.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

Proof
Proof.
Suppose that there exists a continuous surjection
f : K → L0 × L1 . Let g0 , g1 be its component functions, i.e.
f (t) = hg0 (t), g1 (t)i for all t ∈ K .
Let N be a countable elementary submodel of H(θ) for somesufficiently large regular cardinal θ with K , L0 , L1 , f ∈ N
Let hxi |i < ωi be a sequence of distinct elements of L0 ∩ N andy ∈ J(L1 , N). Since f is surjective, for each i < ω, there existti ∈ K such that f (ti ) = hxi , y i.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

Proof
Proof.
Suppose that there exists a continuous surjection
f : K → L0 × L1 . Let g0 , g1 be its component functions, i.e.
f (t) = hg0 (t), g1 (t)i for all t ∈ K .
Let N be a countable elementary submodel of H(θ) for somesufficiently large regular cardinal θ with K , L0 , L1 , f ∈ N
Let hxi |i < ωi be a sequence of distinct elements of L0 ∩ N andy ∈ J(L1 , N). Since f is surjective, for each i < ω, there existti ∈ K such that f (ti ) = hxi , y i.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

Proof (2)Proof (Cont.)
Let i < ω.
Since g1 (ti ) = y ∈ J(L1 , N), we have ti ∈ J(K , N) unlessti < inf(K ∩ N) or ti > (K ∩ N).
Since g0 (ti ) = xi ∈ N, g0  I(ti ) is constant.
Since g1 (ti ) = y ∈ J(L1 , N), g1 → I(ti ) = I(y ).
Let y 0 ∈ I(y ) with y 0 6= y . Then, there exists ti0 ∈ I(ti ) such thatg1 (ti0 ) = y 0 . Thus, f (ti0 ) = hxi , y 0 i.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

Proof (2)Proof (Cont.)
Let i < ω.
Since g1 (ti ) = y ∈ J(L1 , N), we have ti ∈ J(K , N) unlessti < inf(K ∩ N) or ti > (K ∩ N).
Since g0 (ti ) = xi ∈ N, g0  I(ti ) is constant.
Since g1 (ti ) = y ∈ J(L1 , N), g1 → I(ti ) = I(y ).
Let y 0 ∈ I(y ) with y 0 6= y . Then, there exists ti0 ∈ I(ti ) such thatg1 (ti0 ) = y 0 . Thus, f (ti0 ) = hxi , y 0 i.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

Proof (2)Proof (Cont.)
Let i < ω.
Since g1 (ti ) = y ∈ J(L1 , N), we have ti ∈ J(K , N) unlessti < inf(K ∩ N) or ti > (K ∩ N).
Since g0 (ti ) = xi ∈ N, g0  I(ti ) is constant.
Since g1 (ti ) = y ∈ J(L1 , N), g1 → I(ti ) = I(y ).
Let y 0 ∈ I(y ) with y 0 6= y . Then, there exists ti0 ∈ I(ti ) such thatg1 (ti0 ) = y 0 . Thus, f (ti0 ) = hxi , y 0 i.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

Proof (2)Proof (Cont.)
Let i < ω.
Since g1 (ti ) = y ∈ J(L1 , N), we have ti ∈ J(K , N) unlessti < inf(K ∩ N) or ti > (K ∩ N).
Since g0 (ti ) = xi ∈ N, g0  I(ti ) is constant.
Since g1 (ti ) = y ∈ J(L1 , N), g1 → I(ti ) = I(y ).
Let y 0 ∈ I(y ) with y 0 6= y . Then, there exists ti0 ∈ I(ti ) such thatg1 (ti0 ) = y 0 . Thus, f (ti0 ) = hxi , y 0 i.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

Proof (2)Proof (Cont.)
Let i < ω.
Since g1 (ti ) = y ∈ J(L1 , N), we have ti ∈ J(K , N) unlessti < inf(K ∩ N) or ti > (K ∩ N).
Since g0 (ti ) = xi ∈ N, g0  I(ti ) is constant.
Since g1 (ti ) = y ∈ J(L1 , N), g1 → I(ti ) = I(y ).
Let y 0 ∈ I(y ) with y 0 6= y . Then, there exists ti0 ∈ I(ti ) such thatg1 (ti0 ) = y 0 . Thus, f (ti0 ) = hxi , y 0 i.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

Proof (2)Proof (Cont.)
Let i < ω.
Since g1 (ti ) = y ∈ J(L1 , N), we have ti ∈ J(K , N) unlessti < inf(K ∩ N) or ti > (K ∩ N).
Since g0 (ti ) = xi ∈ N, g0  I(ti ) is constant.
Since g1 (ti ) = y ∈ J(L1 , N), g1 → I(ti ) = I(y ).
Let y 0 ∈ I(y ) with y 0 6= y . Then, there exists ti0 ∈ I(ti ) such thatg1 (ti0 ) = y 0 . Thus, f (ti0 ) = hxi , y 0 i.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

Proof (3)Proof (Cont.)
Since K is compact, there is a limit point t of { ti | i < ω }.
Clearly, t is also a limit point of hti0 |i < ωi.
Then,
g1 (t) = lim g1 (ti ) = yi→∞andg1 (t) = lim g1 (ti0 ) = y 0i→∞Since y 6= y 0 , this is a contradiction.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

Proof (3)Proof (Cont.)
Since K is compact, there is a limit point t of { ti | i < ω }.
Clearly, t is also a limit point of hti0 |i < ωi.
Then,
g1 (t) = lim g1 (ti ) = yi→∞andg1 (t) = lim g1 (ti0 ) = y 0i→∞Since y 6= y 0 , this is a contradiction.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves

Proof (3)Proof (Cont.)
Since K is compact, there is a limit point t of { ti | i < ω }.
Clearly, t is also a limit point of hti0 |i < ωi.
Then,
g1 (t) = lim g1 (ti ) = yi→∞andg1 (t) = lim g1 (ti0 ) = y 0i→∞Since y 6= y 0 , this is a contradiction.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves♪If I could draw Peano...

We can prove the following easy corollary:
Theorem
Let K be a compact connected LOTS. Then, there exists a
continuous surjection from K onto K × K if and only if K is
separable, i.e. K is homeomorphic to [0, 1].
So, this is a characterization of the unit interval.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves♪If I could draw Peano...

We can prove the following easy corollary:
Theorem
Let K be a compact connected LOTS. Then, there exists a
continuous surjection from K onto K × K if and only if K is
separable, i.e. K is homeomorphic to [0, 1].
So, this is a characterization of the unit interval.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Space-filling curves♪If I could draw Peano...

We can prove the following easy corollary:
Theorem
Let K be a compact connected LOTS. Then, there exists a
continuous surjection from K onto K × K if and only if K is
separable, i.e. K is homeomorphic to [0, 1].
So, this is a characterization of the unit interval.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

Outline
1

2

3
4

5

Introduction
What are the reals?
Nowhere separable LOTS
Coordinate-wise theorem
Statement
Ideas
Isn’t it nice?
Space-filling curves
Space-filling curves
Linearly Ordered Semigroups
Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups
Open Problems
Open Problems
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

A corollary of the Coordinate-wise theorem
As a corollary of the Coordinate-wise theorem, we can prove
the following:
Definition
Let S be a semigroup. We say that S is cancellative if and only
if for all a, b, c ∈ S, ac = bc implies a = b and ca = cb implies
a = b.
Corollary
Let S be a cancellative topological semigroup that is aconnected LOTS as a topological space. Then, S is separable.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

A corollary of the Coordinate-wise theorem
As a corollary of the Coordinate-wise theorem, we can prove
the following:
Definition
Let S be a semigroup. We say that S is cancellative if and only
if for all a, b, c ∈ S, ac = bc implies a = b and ca = cb implies
a = b.
Corollary
Let S be a cancellative topological semigroup that is aconnected LOTS as a topological space. Then, S is separable.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

A corollary of the Coordinate-wise theorem
As a corollary of the Coordinate-wise theorem, we can prove
the following:
Definition
Let S be a semigroup. We say that S is cancellative if and only
if for all a, b, c ∈ S, ac = bc implies a = b and ca = cb implies
a = b.
Corollary
Let S be a cancellative topological semigroup that is aconnected LOTS as a topological space. Then, S is separable.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

Proof

Proof.
Let S be a cancellative topological semigroup that is aconnected LOTS. Define f : S × S → S × S byf (a, b) = ha, abiThen, f is not coordinate-wise. Thus, S is not nowhereseparable. From this, we can prove that S is separable.
But a stronger result was known long before this result.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

Proof

Proof.
Let S be a cancellative topological semigroup that is aconnected LOTS. Define f : S × S → S × S byf (a, b) = ha, abiThen, f is not coordinate-wise. Thus, S is not nowhereseparable. From this, we can prove that S is separable.
But a stronger result was known long before this result.

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

The theorem of Aczél, Craigen, and Páles
The following theorem was first proved by J. Aczél in 1949, andthe proof was simplified by R. Craigen and Z. Páles in 1989.
Definition
A topological semigroup that is a LOTS as a topological spaceis called a linearly ordered topological semigroup.
Theorem (J. Aczel(1949))
Let S be a connected linearly ordered topological semigroup.
Then, S is order- and semigroup-isomorphic to a subsemigroupof (R, +).
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

The theorem of Aczél, Craigen, and Páles
The following theorem was first proved by J. Aczél in 1949, andthe proof was simplified by R. Craigen and Z. Páles in 1989.
Definition
A topological semigroup that is a LOTS as a topological spaceis called a linearly ordered topological semigroup.
Theorem (J. Aczel(1949))
Let S be a connected linearly ordered topological semigroup.
Then, S is order- and semigroup-isomorphic to a subsemigroupof (R, +).
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

The theorem of Aczél, Craigen, and Páles
The following theorem was first proved by J. Aczél in 1949, andthe proof was simplified by R. Craigen and Z. Páles in 1989.
Definition
A topological semigroup that is a LOTS as a topological spaceis called a linearly ordered topological semigroup.
Theorem (J. Aczel(1949))
Let S be a connected linearly ordered topological semigroup.
Then, S is order- and semigroup-isomorphic to a subsemigroupof (R, +).
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

Outline
1

2

3
4

5

Introduction
What are the reals?
Nowhere separable LOTS
Coordinate-wise theorem
Statement
Ideas
Isn’t it nice?
Space-filling curves
Space-filling curves
Linearly Ordered Semigroups
Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups
Open Problems
Open Problems
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

Linearly ordered semigroups
In fact, semigroups on linearly ordered sets have been studied
so much, but often they are not linearly ordered topologicalsemigroups.
Definition
Let S be a set with a semigroup operation · and a linear
ordering ≤. We say that S is a linearly ordered semigroup if andonly if for all a, b, c ∈ S, a ≤ b implies ac ≤ bc and ca ≤ cb.
These two notions are independent from each other. Namely,
there exists a linearly ordered semigroup that is not linearlyordered topological semigroup and vice versa.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

Linearly ordered semigroups
In fact, semigroups on linearly ordered sets have been studied
so much, but often they are not linearly ordered topologicalsemigroups.
Definition
Let S be a set with a semigroup operation · and a linear
ordering ≤. We say that S is a linearly ordered semigroup if andonly if for all a, b, c ∈ S, a ≤ b implies ac ≤ bc and ca ≤ cb.
These two notions are independent from each other. Namely,
there exists a linearly ordered semigroup that is not linearlyordered topological semigroup and vice versa.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

Linearly ordered semigroups
In fact, semigroups on linearly ordered sets have been studied
so much, but often they are not linearly ordered topologicalsemigroups.
Definition
Let S be a set with a semigroup operation · and a linear
ordering ≤. We say that S is a linearly ordered semigroup if andonly if for all a, b, c ∈ S, a ≤ b implies ac ≤ bc and ca ≤ cb.
These two notions are independent from each other. Namely,
there exists a linearly ordered semigroup that is not linearlyordered topological semigroup and vice versa.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

Positively ordered
There are several results that says that S is order- andsemigroup-isomorhpic to a subsemigroup of (R, +, ≤). By theway, we get the commutativity as a conclusion.
Definition
Let S be a linearly ordered semigroup. We say that S is
positively ordered iff ab ≥ a and ab ≥ b for all a, b ∈ S.

Example([0, ∞), +, ≤) is positively ordered, but (R, +, ≤) is not
positively ordered.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

Positively ordered
There are several results that says that S is order- andsemigroup-isomorhpic to a subsemigroup of (R, +, ≤). By theway, we get the commutativity as a conclusion.
Definition
Let S be a linearly ordered semigroup. We say that S is
positively ordered iff ab ≥ a and ab ≥ b for all a, b ∈ S.

Example([0, ∞), +, ≤) is positively ordered, but (R, +, ≤) is not
positively ordered.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

Positively ordered
There are several results that says that S is order- andsemigroup-isomorhpic to a subsemigroup of (R, +, ≤). By theway, we get the commutativity as a conclusion.
Definition
Let S be a linearly ordered semigroup. We say that S is
positively ordered iff ab ≥ a and ab ≥ b for all a, b ∈ S.

Example([0, ∞), +, ≤) is positively ordered, but (R, +, ≤) is not
positively ordered.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

Positively ordered
There are several results that says that S is order- andsemigroup-isomorhpic to a subsemigroup of (R, +, ≤). By theway, we get the commutativity as a conclusion.
Definition
Let S be a linearly ordered semigroup. We say that S is
positively ordered iff ab ≥ a and ab ≥ b for all a, b ∈ S.

Example([0, ∞), +, ≤) is positively ordered, but (R, +, ≤) is not
positively ordered.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

Archimedean and anomalous pairs
In this talk, when we say S is positively ordered, we shall
implicitly assume S is linearly ordered.
Definition
Let S be a linearly ordered semigroup that is positively ordered.
S is archimedean iff for every a, b ∈ S, whenever a is not
an identity, there exists n ∈ N such that an ≥ b.
a, b ∈ S form an anomalous pair iff a 6= b, and for all
positive natural number n, an < bn+1 and bn < an+1 .

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

Archimedean and anomalous pairs
In this talk, when we say S is positively ordered, we shall
implicitly assume S is linearly ordered.
Definition
Let S be a linearly ordered semigroup that is positively ordered.
S is archimedean iff for every a, b ∈ S, whenever a is not
an identity, there exists n ∈ N such that an ≥ b.
a, b ∈ S form an anomalous pair iff a 6= b, and for all
positive natural number n, an < bn+1 and bn < an+1 .

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

ExampleExample([0, ∞), +) is a positively ordered archimedean semigroup thathas no anomalous pair.
What does a positively ordered semigroup have anomalous
pairs?

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

ExampleExample([0, ∞), +) is a positively ordered archimedean semigroup thathas no anomalous pair.
What does a positively ordered semigroup have anomalous
pairs?

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

Example of anomalous pairs
ExampleDefine a linearly ordered semigroup S = (0, ∞) × [0, ∞) by
ha, xi · hb, y i = ha + b, x + y i and letting S be ordered by thelexicographical ordering.
It is easy to see that S is a linearly ordered semigroup that is
positively ordered and archimedean.
For every a ∈ (0, ∞) and x 6= y ∈ [0, ∞), ha, xi and ha, y i form
an anomalous pair.
Note that [0, ∞) may be replaced by any linearly orderedsemigroup.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

Example of anomalous pairs
ExampleDefine a linearly ordered semigroup S = (0, ∞) × [0, ∞) by
ha, xi · hb, y i = ha + b, x + y i and letting S be ordered by thelexicographical ordering.
It is easy to see that S is a linearly ordered semigroup that is
positively ordered and archimedean.
For every a ∈ (0, ∞) and x 6= y ∈ [0, ∞), ha, xi and ha, y i form
an anomalous pair.
Note that [0, ∞) may be replaced by any linearly orderedsemigroup.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

Example of anomalous pairs
ExampleDefine a linearly ordered semigroup S = (0, ∞) × [0, ∞) by
ha, xi · hb, y i = ha + b, x + y i and letting S be ordered by thelexicographical ordering.
It is easy to see that S is a linearly ordered semigroup that is
positively ordered and archimedean.
For every a ∈ (0, ∞) and x 6= y ∈ [0, ∞), ha, xi and ha, y i form
an anomalous pair.
Note that [0, ∞) may be replaced by any linearly orderedsemigroup.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

Example of anomalous pairs
ExampleDefine a linearly ordered semigroup S = (0, ∞) × [0, ∞) by
ha, xi · hb, y i = ha + b, x + y i and letting S be ordered by thelexicographical ordering.
It is easy to see that S is a linearly ordered semigroup that is
positively ordered and archimedean.
For every a ∈ (0, ∞) and x 6= y ∈ [0, ∞), ha, xi and ha, y i form
an anomalous pair.
Note that [0, ∞) may be replaced by any linearly orderedsemigroup.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

Fuchs’ Theorem
The following is one of the theorems that give an equivalentcondition for a positively ordered semigroup to be order- andsemigroup-isomorphic to a subsemigroup of ([0, ∞), +).
Theorem (L. Fuchs)
Let S be a positively ordered semigroup. Then, S is order- andsemigroup-isomorphic to a subsemigroup of ([0, ∞), +) if andonly if
S is archimedean,
S has no anomalous pairs, andS has no maximal element unless S is a singleton.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

Fuchs’ Theorem
The following is one of the theorems that give an equivalentcondition for a positively ordered semigroup to be order- andsemigroup-isomorphic to a subsemigroup of ([0, ∞), +).
Theorem (L. Fuchs)
Let S be a positively ordered semigroup. Then, S is order- andsemigroup-isomorphic to a subsemigroup of ([0, ∞), +) if andonly if
S is archimedean,
S has no anomalous pairs, andS has no maximal element unless S is a singleton.
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

What if it has anomalous pairs?
In case that S has an anomalous pair, I believe I proved thefollowing.
Conjecture
Let S be a positively ordered archimedean semigroup with no
maximal element. Define an equivalence relation ∼ on S bya ∼ b if and only if either a = b or a and b form an anomalouspair.
If we naturally define S/∼, then S/∼ is order- andsemigroup-isomorphic to a subsemigroup of ([0, ∞), +).

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

What if it has anomalous pairs?
In case that S has an anomalous pair, I believe I proved thefollowing.
Conjecture
Let S be a positively ordered archimedean semigroup with no
maximal element. Define an equivalence relation ∼ on S bya ∼ b if and only if either a = b or a and b form an anomalouspair.
If we naturally define S/∼, then S/∼ is order- andsemigroup-isomorphic to a subsemigroup of ([0, ∞), +).

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

What if it has anomalous pairs?
In case that S has an anomalous pair, I believe I proved thefollowing.
Conjecture
Let S be a positively ordered archimedean semigroup with no
maximal element. Define an equivalence relation ∼ on S bya ∼ b if and only if either a = b or a and b form an anomalouspair.
If we naturally define S/∼, then S/∼ is order- andsemigroup-isomorphic to a subsemigroup of ([0, ∞), +).

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

What if it has anomalous pairs?
In case that S has an anomalous pair, I believe I proved thefollowing.
Conjecture
Let S be a positively ordered archimedean semigroup with no
maximal element. Define an equivalence relation ∼ on S bya ∼ b if and only if either a = b or a and b form an anomalouspair.
If we naturally define S/∼, then S/∼ is order- andsemigroup-isomorphic to a subsemigroup of ([0, ∞), +).

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

So what?
Let S be a positively ordered semigroup. We may define
S-metrizability and S-ultrametrizability of a topological space.
Proposition
Suppose that X is S-metrizable for some positively orderedsemigroup S. If S is archimedean, then X is metrizable.
Question
Suppose that X is S-metrizable for some positively orderedsemigroup S. Then, is X either metrizable or S-ultrametrizable?

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

So what?
Let S be a positively ordered semigroup. We may define
S-metrizability and S-ultrametrizability of a topological space.
Proposition
Suppose that X is S-metrizable for some positively orderedsemigroup S. If S is archimedean, then X is metrizable.
Question
Suppose that X is S-metrizable for some positively orderedsemigroup S. Then, is X either metrizable or S-ultrametrizable?

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups

So what?
Let S be a positively ordered semigroup. We may define
S-metrizability and S-ultrametrizability of a topological space.
Proposition
Suppose that X is S-metrizable for some positively orderedsemigroup S. If S is archimedean, then X is metrizable.
Question
Suppose that X is S-metrizable for some positively orderedsemigroup S. Then, is X either metrizable or S-ultrametrizable?

Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Open Problems

Outline
1

2

3
4

5

Introduction
What are the reals?
Nowhere separable LOTS
Coordinate-wise theorem
Statement
Ideas
Isn’t it nice?
Space-filling curves
Space-filling curves
Linearly Ordered Semigroups
Connected Linearly Ordered Topological Spaces
Linearly ordered semigroups
Open Problems
Open Problems
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Open Problems

Open problems
Question
Can we extend the Coordinate-wise Theorem to more spaces?
To a space in some sense one-dimensional?
Question
Which nowhere separable spaces cannot be ‘rotated’?
Question
Do we really need to assume the compactness of K in the ‘nospace-filling curve’ theorem?
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Open Problems

Open problems
Question
Can we extend the Coordinate-wise Theorem to more spaces?
To a space in some sense one-dimensional?
Question
Which nowhere separable spaces cannot be ‘rotated’?
Question
Do we really need to assume the compactness of K in the ‘nospace-filling curve’ theorem?
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS

Introduction
Coordinate-wise theorem
Space-filling curves
Linearly Ordered Semigroups
Open Problems

Open Problems

Open problems
Question
Can we extend the Coordinate-wise Theorem to more spaces?
To a space in some sense one-dimensional?
Question
Which nowhere separable spaces cannot be ‘rotated’?
Question
Do we really need to assume the compactness of K in the ‘nospace-filling curve’ theorem?
Tetsuya Ishiu

No space-filling curve for a cpt. conn. nowhere sep. LOTS