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Topological homogeneity
Topological homogeneity
Jan van Mill
VU University Amsterdam
Toposym 2011
Topological homogeneity
Outline
1
Introduction
2
Examples of homogeneous compacta
3
Rudin’s Problem
4
van Douwen’s Problem
5
Arhangel skiı̆’s Problem
6
Unique homogeneity
7
Power homogeneity
8
Countable dense homogeneity
Ungar’s Theorems
Actions of groups
The counterexample
9
Open problems
Topological homogeneity
Introduction
Introduction
A topological space X is called (topologically) homogeneous
if for all x, y ∈ X there is a homeomorphism f : X → X such
that f (x) = y.
Topological homogeneity
Introduction
Introduction
A topological space X is called (topologically) homogeneous
if for all x, y ∈ X there is a homeomorphism f : X → X such
that f (x) = y.
Topological homogeneity is not a well understood notion,
especially outside the class of metrizable spaces.
Topological homogeneity
Introduction
Introduction
A topological space X is called (topologically) homogeneous
if for all x, y ∈ X there is a homeomorphism f : X → X such
that f (x) = y.
Topological homogeneity is not a well understood notion,
especially outside the class of metrizable spaces.
There are well developed homeomorphism extension theorems
for manifold-like spaces, both finite- and infinite-dimensional.
Topological homogeneity
Introduction
Introduction
A topological space X is called (topologically) homogeneous
if for all x, y ∈ X there is a homeomorphism f : X → X such
that f (x) = y.
Topological homogeneity is not a well understood notion,
especially outside the class of metrizable spaces.
There are well developed homeomorphism extension theorems
for manifold-like spaces, both finite- and infinite-dimensional.
This means that there is a class of closed subsets Z of such a
space X - usually called Z-sets - with the property that for all
A, B ∈ Z, every homeomorphism f : A → B can be extended
to a homeomorphism f¯: X → X (often with some sort of
control).
Topological homogeneity
Introduction
Introduction
A topological space X is called (topologically) homogeneous
if for all x, y ∈ X there is a homeomorphism f : X → X such
that f (x) = y.
Topological homogeneity is not a well understood notion,
especially outside the class of metrizable spaces.
There are well developed homeomorphism extension theorems
for manifold-like spaces, both finite- and infinite-dimensional.
This means that there is a class of closed subsets Z of such a
space X - usually called Z-sets - with the property that for all
A, B ∈ Z, every homeomorphism f : A → B can be extended
to a homeomorphism f¯: X → X (often with some sort of
control).
Such homeomorphism extension theorems play a crucial role
in the following fundamental characterization theorems:
Topological homogeneity
Introduction
1
2
3
4
Toruńczyk: Hilbert cube and Hilbert space manifolds,
Edwards and Quinn: n-manifolds,
Bestvina: Menger manifolds,
Ageev, Levin, Nagórko: Nöbeling manifolds.
Topological homogeneity
Introduction
1
2
3
4
Toruńczyk: Hilbert cube and Hilbert space manifolds,
Edwards and Quinn: n-manifolds,
Bestvina: Menger manifolds,
Ageev, Levin, Nagórko: Nöbeling manifolds.
So homogeneity works well if not only points are ‘topologically
equivalent’ but instead that all Z-sets are ‘topologically
equivalent’ (in a strong way).
Topological homogeneity
Introduction
1
2
3
4
Toruńczyk: Hilbert cube and Hilbert space manifolds,
Edwards and Quinn: n-manifolds,
Bestvina: Menger manifolds,
Ageev, Levin, Nagórko: Nöbeling manifolds.
So homogeneity works well if not only points are ‘topologically
equivalent’ but instead that all Z-sets are ‘topologically
equivalent’ (in a strong way).
If the homogeneous space under consideration is not close to a
manifold, then many fundamental problems remain unsolved.
Topological homogeneity
Examples of homogeneous compacta
Examples of homogeneous compacta
Compact metrizable spaces:
Topological homogeneity
Examples of homogeneous compacta
Examples of homogeneous compacta
Compact metrizable spaces:
Cantor set, compact manifolds without boundary, Hilbert cube
manifolds, universal Menger continua, pseudoarc, circle of
pseudoarcs, compact (metrizable) groups, solenoidial spaces,
Case continuum, homogeneous but not bihomogeneous
continua of Kuberberg (and others), the homogeneous arcwise
connected non-locally connected curve of Prajs, products, etc.
Topological homogeneity
Examples of homogeneous compacta
Examples of homogeneous compacta
Compact metrizable spaces:
Cantor set, compact manifolds without boundary, Hilbert cube
manifolds, universal Menger continua, pseudoarc, circle of
pseudoarcs, compact (metrizable) groups, solenoidial spaces,
Case continuum, homogeneous but not bihomogeneous
continua of Kuberberg (and others), the homogeneous arcwise
connected non-locally connected curve of Prajs, products, etc.
A nice variety of spaces and very well-developed homogeneity
theories leading to fundamental characterization theorems.
Topological homogeneity
Examples of homogeneous compacta
Compact spaces of uncountable weight:
Topological homogeneity
Examples of homogeneous compacta
Compact spaces of uncountable weight:
compact groups, Tychonoff cubes, Cantor cubes, compact
ordered spaces of Maurice (cellularity c), compact ordered
spaces of van Douwen (countable π-weight), infinite products
of compact zero-dimensional first countable spaces (Motorov;
generalized by Dow and Pearl), solenoidial spaces, Kunen’s
compact homogeneous L-space, Kunen and de la Vega’s
compact homogeneous S-space, examples by de la Vega,
Chatyrko, Fedorchuk, Milovich and van Mill, products, etc.
Topological homogeneity
Examples of homogeneous compacta
Compact spaces of uncountable weight:
compact groups, Tychonoff cubes, Cantor cubes, compact
ordered spaces of Maurice (cellularity c), compact ordered
spaces of van Douwen (countable π-weight), infinite products
of compact zero-dimensional first countable spaces (Motorov;
generalized by Dow and Pearl), solenoidial spaces, Kunen’s
compact homogeneous L-space, Kunen and de la Vega’s
compact homogeneous S-space, examples by de la Vega,
Chatyrko, Fedorchuk, Milovich and van Mill, products, etc.
No new general classes of examples. More ad-hoc
constructions. Instead, many techniques developed for proving
that spaces are NOT homogeneous.
Topological homogeneity
Examples of homogeneous compacta
There are basically two methods for constructing ‘large’
homogeneous compacta:
Topological homogeneity
Examples of homogeneous compacta
There are basically two methods for constructing ‘large’
homogeneous compacta:
Form a large product of first countable zero-dimensional
compacta and apply the Motorov (or Dow and Pearl)
Theorem to conclude it is homogeneous. Multiply this
product by a compact group of large weight.
Topological homogeneity
Examples of homogeneous compacta
There are basically two methods for constructing ‘large’
homogeneous compacta:
Form a large product of first countable zero-dimensional
compacta and apply the Motorov (or Dow and Pearl)
Theorem to conclude it is homogeneous. Multiply this
product by a compact group of large weight.
Ad-hoc methods such as used by Maurice, Fedorchuk,
Chatyrko, van Douwen, Kunen, de la Vega, Milovich and van
Mill.
Topological homogeneity
Examples of homogeneous compacta
There are basically two methods for constructing ‘large’
homogeneous compacta:
Form a large product of first countable zero-dimensional
compacta and apply the Motorov (or Dow and Pearl)
Theorem to conclude it is homogeneous. Multiply this
product by a compact group of large weight.
Ad-hoc methods such as used by Maurice, Fedorchuk,
Chatyrko, van Douwen, Kunen, de la Vega, Milovich and van
Mill.
It is clear that there is still a lot to do.
Topological homogeneity
Examples of homogeneous compacta
There are basically two methods for constructing ‘large’
homogeneous compacta:
Form a large product of first countable zero-dimensional
compacta and apply the Motorov (or Dow and Pearl)
Theorem to conclude it is homogeneous. Multiply this
product by a compact group of large weight.
Ad-hoc methods such as used by Maurice, Fedorchuk,
Chatyrko, van Douwen, Kunen, de la Vega, Milovich and van
Mill.
It is clear that there is still a lot to do.
We review some of the fundamental open problems that
remained unsolved for decades.
Topological homogeneity
Rudin’s Problem
Rudin’s Problem
Topological homogeneity
Rudin’s Problem
Rudin’s Problem
Walter Rudin proved in 1956 that N∗ = βN \ N is not
homogeneous under CH.
Topological homogeneity
Rudin’s Problem
Rudin’s Problem
Walter Rudin proved in 1956 that N∗ = βN \ N is not
homogeneous under CH.
He proved the existence of two types of points in N∗ with
evident different topological behavior: the P-points and the
non-P-points.
Topological homogeneity
Rudin’s Problem
Rudin’s Problem
Walter Rudin proved in 1956 that N∗ = βN \ N is not
homogeneous under CH.
He proved the existence of two types of points in N∗ with
evident different topological behavior: the P-points and the
non-P-points.
Shelah proved in 1977 or 1978 that P-points need not exist in
N∗ (published in 1982).
Topological homogeneity
Rudin’s Problem
Rudin’s Problem
Walter Rudin proved in 1956 that N∗ = βN \ N is not
homogeneous under CH.
He proved the existence of two types of points in N∗ with
evident different topological behavior: the P-points and the
non-P-points.
Shelah proved in 1977 or 1978 that P-points need not exist in
N∗ (published in 1982).
In 1967, Frolı́k established the inhomogeneity of N∗ in ZFC;
moreover, he showed that N∗ decomposes into 2c equivalence
classes under homeomorphisms.
Topological homogeneity
Rudin’s Problem
Rudin’s Problem
Walter Rudin proved in 1956 that N∗ = βN \ N is not
homogeneous under CH.
He proved the existence of two types of points in N∗ with
evident different topological behavior: the P-points and the
non-P-points.
Shelah proved in 1977 or 1978 that P-points need not exist in
N∗ (published in 1982).
In 1967, Frolı́k established the inhomogeneity of N∗ in ZFC;
moreover, he showed that N∗ decomposes into 2c equivalence
classes under homeomorphisms.
His proof is based on cardinality considerations and does not
yield points with obvious different topological behavior.
Topological homogeneity
Rudin’s Problem
Rudin’s Problem
Walter Rudin proved in 1956 that N∗ = βN \ N is not
homogeneous under CH.
He proved the existence of two types of points in N∗ with
evident different topological behavior: the P-points and the
non-P-points.
Shelah proved in 1977 or 1978 that P-points need not exist in
N∗ (published in 1982).
In 1967, Frolı́k established the inhomogeneity of N∗ in ZFC;
moreover, he showed that N∗ decomposes into 2c equivalence
classes under homeomorphisms.
His proof is based on cardinality considerations and does not
yield points with obvious different topological behavior.
Eleven years later, in 1978, Kunen proved the existence of two
types of points in N∗ with evident different topological
behavior: the weak P-points and the non-weak-P-points.
Topological homogeneity
Rudin’s Problem
in 1958, Rudin returned to N∗ , and asked whether the
inhomogeneity of N∗ was a consequence of the fact that it
contains no nontrivial convergent sequences.
Topological homogeneity
Rudin’s Problem
in 1958, Rudin returned to N∗ , and asked whether the
inhomogeneity of N∗ was a consequence of the fact that it
contains no nontrivial convergent sequences.
This problem has been open for more than half of a century
now, and is known as Rudin’s Problem.
Problem
Does every infinite homogeneous compact space contain a
nontrivial convergent sequence?
Topological homogeneity
Rudin’s Problem
in 1958, Rudin returned to N∗ , and asked whether the
inhomogeneity of N∗ was a consequence of the fact that it
contains no nontrivial convergent sequences.
This problem has been open for more than half of a century
now, and is known as Rudin’s Problem.
Problem
Does every infinite homogeneous compact space contain a
nontrivial convergent sequence?
YES for compact groups (Kuz minov, Ivanovskij), and more
generally, dyadic compacta. YES for first countable compacta
(trivial).
Topological homogeneity
Rudin’s Problem
in 1958, Rudin returned to N∗ , and asked whether the
inhomogeneity of N∗ was a consequence of the fact that it
contains no nontrivial convergent sequences.
This problem has been open for more than half of a century
now, and is known as Rudin’s Problem.
Problem
Does every infinite homogeneous compact space contain a
nontrivial convergent sequence?
YES for compact groups (Kuz minov, Ivanovskij), and more
generally, dyadic compacta. YES for first countable compacta
(trivial).
Even unknown for separable compacta, or for compacta of
countable π-weight.
Topological homogeneity
van Douwen’s Problem
van Douwen’s Problem
Topological homogeneity
van Douwen’s Problem
van Douwen’s Problem
Haar measure on a compact group clearly implies that is has
countable cellularity.
Topological homogeneity
van Douwen’s Problem
van Douwen’s Problem
Haar measure on a compact group clearly implies that is has
countable cellularity.
Are there homogeneous compacta of uncountable cellularity?
Topological homogeneity
van Douwen’s Problem
van Douwen’s Problem
Haar measure on a compact group clearly implies that is has
countable cellularity.
Are there homogeneous compacta of uncountable cellularity?
YES, the ordered compacta of Maurice are homogeneous and
have cellularity c.
Topological homogeneity
van Douwen’s Problem
van Douwen’s Problem
Haar measure on a compact group clearly implies that is has
countable cellularity.
Are there homogeneous compacta of uncountable cellularity?
YES, the ordered compacta of Maurice are homogeneous and
have cellularity c.
The infinite product of the Alexandroff duplicate of the
Cantor set is another example by the Motorov Theorem.
Topological homogeneity
van Douwen’s Problem
van Douwen’s Problem
Haar measure on a compact group clearly implies that is has
countable cellularity.
Are there homogeneous compacta of uncountable cellularity?
YES, the ordered compacta of Maurice are homogeneous and
have cellularity c.
The infinite product of the Alexandroff duplicate of the
Cantor set is another example by the Motorov Theorem.
Large products of first countable zero-dimensional compacta
and compact groups have cellularity at most c.
Topological homogeneity
van Douwen’s Problem
van Douwen’s Problem
Haar measure on a compact group clearly implies that is has
countable cellularity.
Are there homogeneous compacta of uncountable cellularity?
YES, the ordered compacta of Maurice are homogeneous and
have cellularity c.
The infinite product of the Alexandroff duplicate of the
Cantor set is another example by the Motorov Theorem.
Large products of first countable zero-dimensional compacta
and compact groups have cellularity at most c.
Problem
Is there a homogeneous compact space with cellularity greater
than c?
Topological homogeneity
van Douwen’s Problem
van Douwen’s Problem
Haar measure on a compact group clearly implies that is has
countable cellularity.
Are there homogeneous compacta of uncountable cellularity?
YES, the ordered compacta of Maurice are homogeneous and
have cellularity c.
The infinite product of the Alexandroff duplicate of the
Cantor set is another example by the Motorov Theorem.
Large products of first countable zero-dimensional compacta
and compact groups have cellularity at most c.
Problem
Is there a homogeneous compact space with cellularity greater
than c?
This has been open now for about 30 years, and is known as
van Douwen’s Problem.
Topological homogeneity
van Douwen’s Problem
Related problems are:
Problem
Is every compact compact space a continuous image of a
homogeneous compact space?
Problem
Is there for every compact space X a compact space Y such that
X × Y is homogeneous?
Topological homogeneity
van Douwen’s Problem
Related problems are:
Problem
Is every compact compact space a continuous image of a
homogeneous compact space?
Problem
Is there for every compact space X a compact space Y such that
X × Y is homogeneous?
Motorov has shown that the familiar sin x1 -continuum in the
plane is not a retract of a homogenous compact space.
Topological homogeneity
van Douwen’s Problem
Related problems are:
Problem
Is every compact compact space a continuous image of a
homogeneous compact space?
Problem
Is there for every compact space X a compact space Y such that
X × Y is homogeneous?
Motorov has shown that the familiar sin x1 -continuum in the
plane is not a retract of a homogenous compact space.
Farah proved that if X is a compact βN-space, then X × Y is
inhomogeneous for every compact space Y .
Topological homogeneity
Arhangel skiı̆’s Problem
Arhangel skiı̆’s Problem
Topological homogeneity
Arhangel skiı̆’s Problem
Arhangel skiı̆’s Problem
Ismael and Hart and Kunen observed that |X| = 2χ(X) for
every homogeneous compactum.
Topological homogeneity
Arhangel skiı̆’s Problem
Arhangel skiı̆’s Problem
Ismael and Hart and Kunen observed that |X| = 2χ(X) for
every homogeneous compactum.
Hence a homogeneous compact space of cardinality at most c
is first countable under CH.
Topological homogeneity
Arhangel skiı̆’s Problem
Arhangel skiı̆’s Problem
Ismael and Hart and Kunen observed that |X| = 2χ(X) for
every homogeneous compactum.
Hence a homogeneous compact space of cardinality at most c
is first countable under CH.
De la Vega proved that if X is a homogeneous compact space
then |X| ≤ 2t(X) .
Topological homogeneity
Arhangel skiı̆’s Problem
Arhangel skiı̆’s Problem
Ismael and Hart and Kunen observed that |X| = 2χ(X) for
every homogeneous compactum.
Hence a homogeneous compact space of cardinality at most c
is first countable under CH.
De la Vega proved that if X is a homogeneous compact space
then |X| ≤ 2t(X) .
Hence a homogeneous compact space of countable tightness
is first countable under CH.
Topological homogeneity
Arhangel skiı̆’s Problem
Arhangel skiı̆’s Problem
Ismael and Hart and Kunen observed that |X| = 2χ(X) for
every homogeneous compactum.
Hence a homogeneous compact space of cardinality at most c
is first countable under CH.
De la Vega proved that if X is a homogeneous compact space
then |X| ≤ 2t(X) .
Hence a homogeneous compact space of countable tightness
is first countable under CH.
Arhangel skiı̆’s Problem asks whether this is true in ZFC.
Problem
Is every homogeneous compact space of countable tightness first
countable?
Topological homogeneity
Arhangel skiı̆’s Problem
Arhangel skiı̆’s Problem
Ismael and Hart and Kunen observed that |X| = 2χ(X) for
every homogeneous compactum.
Hence a homogeneous compact space of cardinality at most c
is first countable under CH.
De la Vega proved that if X is a homogeneous compact space
then |X| ≤ 2t(X) .
Hence a homogeneous compact space of countable tightness
is first countable under CH.
Arhangel skiı̆’s Problem asks whether this is true in ZFC.
Problem
Is every homogeneous compact space of countable tightness first
countable?
This has been posed at the third Prague Topological
Symposium in 1971. So it open now for more than 40 years.
Topological homogeneity
Arhangel skiı̆’s Problem
In most of these problems, it is known that compactness is
essential.
Topological homogeneity
Arhangel skiı̆’s Problem
In most of these problems, it is known that compactness is
essential.
There is a countably compact homogeneous extremally
disconnected space (Comfort and van Mill).
Topological homogeneity
Arhangel skiı̆’s Problem
In most of these problems, it is known that compactness is
essential.
There is a countably compact homogeneous extremally
disconnected space (Comfort and van Mill).
Every compact space embeds as a retract in a countably
compact homogeneous space (Okromeshko, Comfort and van
Mill).
Topological homogeneity
Arhangel skiı̆’s Problem
In most of these problems, it is known that compactness is
essential.
There is a countably compact homogeneous extremally
disconnected space (Comfort and van Mill).
Every compact space embeds as a retract in a countably
compact homogeneous space (Okromeshko, Comfort and van
Mill).
There is a countably compact homogeneous space of
countably tightness which is not first countable (just take the
standard Σ-product in 2ω1 ).
Topological homogeneity
Unique homogeneity
A space X is called uniquely homogeneous provided that for
all x, y ∈ X there is a unique homeomorphism moving x to y.
So a uniquely homogeneous space is a space which is ‘barely’
homogeneous.
Topological homogeneity
Unique homogeneity
A space X is called uniquely homogeneous provided that for
all x, y ∈ X there is a unique homeomorphism moving x to y.
So a uniquely homogeneous space is a space which is ‘barely’
homogeneous.
A space X is uniquely homogeneous if and only if every
homeomorphism of X with a fixed-point is the identity.
Topological homogeneity
Unique homogeneity
A space X is called uniquely homogeneous provided that for
all x, y ∈ X there is a unique homeomorphism moving x to y.
So a uniquely homogeneous space is a space which is ‘barely’
homogeneous.
A space X is uniquely homogeneous if and only if every
homeomorphism of X with a fixed-point is the identity.
This concept is due to Burgess who asked in 1955 whether
there exists a non-trivial uniquely homogeneous metrizable
continuum.
Topological homogeneity
Unique homogeneity
A space X is called uniquely homogeneous provided that for
all x, y ∈ X there is a unique homeomorphism moving x to y.
So a uniquely homogeneous space is a space which is ‘barely’
homogeneous.
A space X is uniquely homogeneous if and only if every
homeomorphism of X with a fixed-point is the identity.
This concept is due to Burgess who asked in 1955 whether
there exists a non-trivial uniquely homogeneous metrizable
continuum.
Theorem (Barit and Renaud, 1978)
If X is locally compact, metrizable and uniquely homogeneous
then |X| ≤ 2.
Topological homogeneity
Unique homogeneity
A space X is called uniquely homogeneous provided that for
all x, y ∈ X there is a unique homeomorphism moving x to y.
So a uniquely homogeneous space is a space which is ‘barely’
homogeneous.
A space X is uniquely homogeneous if and only if every
homeomorphism of X with a fixed-point is the identity.
This concept is due to Burgess who asked in 1955 whether
there exists a non-trivial uniquely homogeneous metrizable
continuum.
Theorem (Barit and Renaud, 1978)
If X is locally compact, metrizable and uniquely homogeneous
then |X| ≤ 2.
The proof of this theorem is based on the Effros Open
Mapping Principle in actions of Polish groups on Polish
spaces.
Topological homogeneity
Unique homogeneity
There is a nontrivial connected and locally connnected
separable metrizable topological group having no
homeomorphisms other than translations (van Mill, 1983).
This space is uniquely homogeneous.
Topological homogeneity
Unique homogeneity
There is a nontrivial connected and locally connnected
separable metrizable topological group having no
homeomorphisms other than translations (van Mill, 1983).
This space is uniquely homogeneous.
Problem
Is there a Polish uniquely homogeneous space?
Topological homogeneity
Unique homogeneity
There is a nontrivial connected and locally connnected
separable metrizable topological group having no
homeomorphisms other than translations (van Mill, 1983).
This space is uniquely homogeneous.
Problem
Is there a Polish uniquely homogeneous space?
Recent work on uniquely homogeneous spaces was done by
Arhangel skiı̆ and van Mill. They proved:
Topological homogeneity
Unique homogeneity
There is a nontrivial connected and locally connnected
separable metrizable topological group having no
homeomorphisms other than translations (van Mill, 1983).
This space is uniquely homogeneous.
Problem
Is there a Polish uniquely homogeneous space?
Recent work on uniquely homogeneous spaces was done by
Arhangel skiı̆ and van Mill. They proved:
Theorem
Every infinite uniquely homogeneous space is connected. An
infinite ordered space is not uniquely homogeneous.
Topological homogeneity
Unique homogeneity
Theorem
There is a uniquely homogeneous space which contains a copy of
the Cantor cube of weight 2c .
Topological homogeneity
Unique homogeneity
Theorem
There is a uniquely homogeneous space which contains a copy of
the Cantor cube of weight 2c .
Hence βN is a subspace of a uniquely homogeneous space.
Topological homogeneity
Unique homogeneity
Theorem
There is a uniquely homogeneous space which contains a copy of
the Cantor cube of weight 2c .
Hence βN is a subspace of a uniquely homogeneous space.
Problem
Is there a compact uniquely homogeneous space?
Topological homogeneity
Unique homogeneity
Theorem
There is a uniquely homogeneous space which contains a copy of
the Cantor cube of weight 2c .
Hence βN is a subspace of a uniquely homogeneous space.
Problem
Is there a compact uniquely homogeneous space?
Problem
Are there uniquely homogeneous spaces of arbitrarily large weight?
Topological homogeneity
Unique homogeneity
Theorem
There is a uniquely homogeneous space which contains a copy of
the Cantor cube of weight 2c .
Hence βN is a subspace of a uniquely homogeneous space.
Problem
Is there a compact uniquely homogeneous space?
Problem
Are there uniquely homogeneous spaces of arbitrarily large weight?
Problem
Is every compact space a subspace of some uniquely homogeneous
space?
Topological homogeneity
Unique homogeneity
Let X be uniquely homogeneous and fix an element e ∈ X.
For every x ∈ X let fx be the unique homeomorphism taking
e onto x.
Define a binary operation ’·’ and an operation ’−1 ’ on X by
x·y = fx (y),
x−1 = fx−1 (e).
Topological homogeneity
Unique homogeneity
Let X be uniquely homogeneous and fix an element e ∈ X.
For every x ∈ X let fx be the unique homeomorphism taking
e onto x.
Define a binary operation ’·’ and an operation ’−1 ’ on X by
x·y = fx (y),
x−1 = fx−1 (e).
It is easy to see that this makes X into a left topological
group. That is, ’·’ is a group operation on X, and all left
translations of X are homeomorphisms.
Topological homogeneity
Unique homogeneity
Let X be uniquely homogeneous and fix an element e ∈ X.
For every x ∈ X let fx be the unique homeomorphism taking
e onto x.
Define a binary operation ’·’ and an operation ’−1 ’ on X by
x·y = fx (y),
x−1 = fx−1 (e).
It is easy to see that this makes X into a left topological
group. That is, ’·’ is a group operation on X, and all left
translations of X are homeomorphisms.
It is natural to ask whether this operation gives X the
structure of a topological group.
Topological homogeneity
Unique homogeneity
Let X be uniquely homogeneous and fix an element e ∈ X.
For every x ∈ X let fx be the unique homeomorphism taking
e onto x.
Define a binary operation ’·’ and an operation ’−1 ’ on X by
x·y = fx (y),
x−1 = fx−1 (e).
It is easy to see that this makes X into a left topological
group. That is, ’·’ is a group operation on X, and all left
translations of X are homeomorphisms.
It is natural to ask whether this operation gives X the
structure of a topological group.
YES if X is locally compact, separable and metrizable (Barit
and Renaud, based on the Effros Theorem). NO for separable
metrizable spaces (van Mill).
Topological homogeneity
Unique homogeneity
Let X be uniquely homogeneous and fix an element e ∈ X.
For every x ∈ X let fx be the unique homeomorphism taking
e onto x.
Define a binary operation ’·’ and an operation ’−1 ’ on X by
x·y = fx (y),
x−1 = fx−1 (e).
It is easy to see that this makes X into a left topological
group. That is, ’·’ is a group operation on X, and all left
translations of X are homeomorphisms.
It is natural to ask whether this operation gives X the
structure of a topological group.
YES if X is locally compact, separable and metrizable (Barit
and Renaud, based on the Effros Theorem). NO for separable
metrizable spaces (van Mill).
What about the structure of a semitopological group? That
is, the group operation is separately continuous. Or a
quasitopological group? That is, a semitopological group such
that the inverse operation is continuous.
Topological homogeneity
Unique homogeneity
A space X is 2-flexible if, for all a, b ∈ X and open
neighborhood O(b) of b, there is an open neighborhood O(a)
of a such that, for any z ∈ O(a), there is a homeomorphism h
of X satisfying the following conditions: h(a) = z and
h(b) ∈ O(b).
Topological homogeneity
Unique homogeneity
A space X is 2-flexible if, for all a, b ∈ X and open
neighborhood O(b) of b, there is an open neighborhood O(a)
of a such that, for any z ∈ O(a), there is a homeomorphism h
of X satisfying the following conditions: h(a) = z and
h(b) ∈ O(b).
A space X will be called Abelian if all homeomorphisms of X
commute pairwise.
Topological homogeneity
Unique homogeneity
A space X is 2-flexible if, for all a, b ∈ X and open
neighborhood O(b) of b, there is an open neighborhood O(a)
of a such that, for any z ∈ O(a), there is a homeomorphism h
of X satisfying the following conditions: h(a) = z and
h(b) ∈ O(b).
A space X will be called Abelian if all homeomorphisms of X
commute pairwise.
A space X will be called skew-2-flexible if, for any a, b in X
and any open neighborhood O(b) of b, there is an open
neighborhood O(a) of a such that, for every z ∈ O(a), there
is a homeomorphism g of X satisfying the following
conditions: g(a) = z and b ∈ g(O(b)).
Topological homogeneity
Unique homogeneity
A space X is 2-flexible if, for all a, b ∈ X and open
neighborhood O(b) of b, there is an open neighborhood O(a)
of a such that, for any z ∈ O(a), there is a homeomorphism h
of X satisfying the following conditions: h(a) = z and
h(b) ∈ O(b).
A space X will be called Abelian if all homeomorphisms of X
commute pairwise.
A space X will be called skew-2-flexible if, for any a, b in X
and any open neighborhood O(b) of b, there is an open
neighborhood O(a) of a such that, for every z ∈ O(a), there
is a homeomorphism g of X satisfying the following
conditions: g(a) = z and b ∈ g(O(b)).
A space X will be called Boolean if every homeomorphism of
X is an involution. That is, a homeomorphism f such that
f ◦ f is the identity.
Topological homogeneity
Unique homogeneity
Theorem (Arhangel skiı̆ and van Mill)
Let X be a uniquely homogeneous space. Then following
statements are equivalent.
1
X is 2-flexible,
2
the standard group structure on X is semitopological,
3
X is homeomorphic to a semitopological group,
4
X is Abelian,
5
6
the standard group structure on X is semitopological and
Abelian,
X is homeomorphic to an Abelian semitopological group.
Topological homogeneity
Unique homogeneity
Theorem (Arhangel skiı̆ and van Mill)
Let X be a uniquely homogeneous space. Then following
statements are equivalent.
1
X is skew-2-flexible,
2
X is 2-flexible and skew-2-flexible,
3
the standard group structure on X is quasitopological,
4
X is homeomorphic to a quasitopological group,
5
X is Boolean,
6
7
the standard group structure on X is quasitopological and
Boolean,
X is homeomorphic to a Boolean quasitopological group.
Topological homogeneity
Unique homogeneity
Hence for uniquely homogeneous spaces, skew-2-flexibility
implies 2-flexibility.
Topological homogeneity
Unique homogeneity
Hence for uniquely homogeneous spaces, skew-2-flexibility
implies 2-flexibility.
There is a homogeneous Polish space which is skew-2-flexible
but not 2-flexible.
Topological homogeneity
Unique homogeneity
Hence for uniquely homogeneous spaces, skew-2-flexibility
implies 2-flexibility.
There is a homogeneous Polish space which is skew-2-flexible
but not 2-flexible.
There is a uniquely homogeneous space which is Abelian but
not Boolean.
Topological homogeneity
Unique homogeneity
Hence for uniquely homogeneous spaces, skew-2-flexibility
implies 2-flexibility.
There is a homogeneous Polish space which is skew-2-flexible
but not 2-flexible.
There is a uniquely homogeneous space which is Abelian but
not Boolean.
Hence there is a uniquely homogeneous space which is
2-flexible but not skew-2-flexible.
Topological homogeneity
Power homogeneity
A space X is called power homogeneous if X μ is
homogeneous for some cardinal number μ.
Topological homogeneity
Power homogeneity
A space X is called power homogeneous if X μ is
homogeneous for some cardinal number μ.
A power homogeneous space need not be homogeneous, as
the unit interval [0, 1] demonstrates.
Topological homogeneity
Power homogeneity
A space X is called power homogeneous if X μ is
homogeneous for some cardinal number μ.
A power homogeneous space need not be homogeneous, as
the unit interval [0, 1] demonstrates.
Theorem (van Douwen, 1978)
If X is power homogeneous, then |X| ≤ 2π(X) .
Topological homogeneity
Power homogeneity
A space X is called power homogeneous if X μ is
homogeneous for some cardinal number μ.
A power homogeneous space need not be homogeneous, as
the unit interval [0, 1] demonstrates.
Theorem (van Douwen, 1978)
If X is power homogeneous, then |X| ≤ 2π(X) .
Results that are in the same spirit were obtained by
Arhangel skiı̆, Ridderbos, de la Vega, van Mill, Milovich,
Carlson, Juhász, Nyikos, Szentmiklóssy and others, mainly in
the area of cardinal functions.
Topological homogeneity
Power homogeneity
Theorem
1 (Arhangel skiı̆, 2002) Every compact scattered power
homogeneous space is countable.
Topological homogeneity
Power homogeneity
Theorem
1 (Arhangel skiı̆, 2002) Every compact scattered power
homogeneous space is countable.
2
(Arhangel skiı̆, 2004) Every power homogeneous, locally
compact and monotonically normal space is first countable.
Topological homogeneity
Power homogeneity
Theorem
1 (Arhangel skiı̆, 2002) Every compact scattered power
homogeneous space is countable.
2
3
(Arhangel skiı̆, 2004) Every power homogeneous, locally
compact and monotonically normal space is first countable.
(Arhangel skiı̆, 2005) If X is a power homogeneous compact
space which is first countable at a dense set of pojnts, then X
is first countable (and hence of cardinality at most c.)
Topological homogeneity
Power homogeneity
Theorem
1 (Arhangel skiı̆, 2002) Every compact scattered power
homogeneous space is countable.
2
3
4
(Arhangel skiı̆, 2004) Every power homogeneous, locally
compact and monotonically normal space is first countable.
(Arhangel skiı̆, 2005) If X is a power homogeneous compact
space which is first countable at a dense set of pojnts, then X
is first countable (and hence of cardinality at most c.)
(Ridderbos, 2006) If X is power homogeneous, then
|X| ≤ 2πχ (X)c(X) .
Topological homogeneity
Power homogeneity
Theorem
1 (Arhangel skiı̆, 2002) Every compact scattered power
homogeneous space is countable.
2
3
4
5
(Arhangel skiı̆, 2004) Every power homogeneous, locally
compact and monotonically normal space is first countable.
(Arhangel skiı̆, 2005) If X is a power homogeneous compact
space which is first countable at a dense set of pojnts, then X
is first countable (and hence of cardinality at most c.)
(Ridderbos, 2006) If X is power homogeneous, then
|X| ≤ 2πχ (X)c(X) .
(Arhangel skiı̆, van Mill, Ridderbos, 2007) If X is power
homogeneous, then |X| ≤ 2t(X) .
Topological homogeneity
Power homogeneity
Theorem
1 (Arhangel skiı̆, 2002) Every compact scattered power
homogeneous space is countable.
2
3
4
5
6
(Arhangel skiı̆, 2004) Every power homogeneous, locally
compact and monotonically normal space is first countable.
(Arhangel skiı̆, 2005) If X is a power homogeneous compact
space which is first countable at a dense set of pojnts, then X
is first countable (and hence of cardinality at most c.)
(Ridderbos, 2006) If X is power homogeneous, then
|X| ≤ 2πχ (X)c(X) .
(Arhangel skiı̆, van Mill, Ridderbos, 2007) If X is power
homogeneous, then |X| ≤ 2t(X) .
(Ridderbos, 2009) If X is a power homogeneous, hereditarily
normal compact space, then |X| ≤ 2c(X) .
Topological homogeneity
Power homogeneity
Theorem
1 (Arhangel skiı̆, 2002) Every compact scattered power
homogeneous space is countable.
2
3
4
5
6
7
(Arhangel skiı̆, 2004) Every power homogeneous, locally
compact and monotonically normal space is first countable.
(Arhangel skiı̆, 2005) If X is a power homogeneous compact
space which is first countable at a dense set of pojnts, then X
is first countable (and hence of cardinality at most c.)
(Ridderbos, 2006) If X is power homogeneous, then
|X| ≤ 2πχ (X)c(X) .
(Arhangel skiı̆, van Mill, Ridderbos, 2007) If X is power
homogeneous, then |X| ≤ 2t(X) .
(Ridderbos, 2009) If X is a power homogeneous, hereditarily
normal compact space, then |X| ≤ 2c(X) .
(Ridderbos, 2007) If X is connected, power homogeneous and
somewhere locally connected, then X is locally connected.
Topological homogeneity
Countable dense homogeneity
Definition
A space X is countable dense homogeneous (abbreviated: CDH) if
given any two countable dense subsets D and E of X there is a
homeomorphism f : X → X such that f (D) = E.
Topological homogeneity
Countable dense homogeneity
Definition
A space X is countable dense homogeneous (abbreviated: CDH) if
given any two countable dense subsets D and E of X there is a
homeomorphism f : X → X such that f (D) = E.
This notion is of interest only if X is separable. Most of the
spaces we are interested in now are both separable and
metrizable.
Topological homogeneity
Countable dense homogeneity
Definition
A space X is countable dense homogeneous (abbreviated: CDH) if
given any two countable dense subsets D and E of X there is a
homeomorphism f : X → X such that f (D) = E.
This notion is of interest only if X is separable. Most of the
spaces we are interested in now are both separable and
metrizable.
The first result in this area is due to Cantor, who showed that
the reals are CDH. Fréchet and Brouwer, independently,
proved that the same is true for the n-dimensional Euclidean
space Rn .
Topological homogeneity
Countable dense homogeneity
Definition
A space X is countable dense homogeneous (abbreviated: CDH) if
given any two countable dense subsets D and E of X there is a
homeomorphism f : X → X such that f (D) = E.
This notion is of interest only if X is separable. Most of the
spaces we are interested in now are both separable and
metrizable.
The first result in this area is due to Cantor, who showed that
the reals are CDH. Fréchet and Brouwer, independently,
proved that the same is true for the n-dimensional Euclidean
space Rn .
In 1962, Fort proved that the Hilbert cube is also CDH.
Topological homogeneity
Countable dense homogeneity
Definition
A space X is countable dense homogeneous (abbreviated: CDH) if
given any two countable dense subsets D and E of X there is a
homeomorphism f : X → X such that f (D) = E.
This notion is of interest only if X is separable. Most of the
spaces we are interested in now are both separable and
metrizable.
The first result in this area is due to Cantor, who showed that
the reals are CDH. Fréchet and Brouwer, independently,
proved that the same is true for the n-dimensional Euclidean
space Rn .
In 1962, Fort proved that the Hilbert cube is also CDH.
There are many other CDH-spaces, as the following results
show.
Topological homogeneity
Countable dense homogeneity
Definition
A space X is called strongly locally homogeneous (abbreviated
SLH) if it has a base B such that for all B ∈ B and x, y ∈ B
there is a homeomorphism f : X → X that is supported on B
(that is, f is the identity outside B) and moves x to y.
Topological homogeneity
Countable dense homogeneity
Definition
A space X is called strongly locally homogeneous (abbreviated
SLH) if it has a base B such that for all B ∈ B and x, y ∈ B
there is a homeomorphism f : X → X that is supported on B
(that is, f is the identity outside B) and moves x to y.
Bessaga and Pelczyński published a paper in 1969 in which
they prove that a Polish SLH space is CDH. This paper was
submitted for publication in February, 1969.
Topological homogeneity
Countable dense homogeneity
Definition
A space X is called strongly locally homogeneous (abbreviated
SLH) if it has a base B such that for all B ∈ B and x, y ∈ B
there is a homeomorphism f : X → X that is supported on B
(that is, f is the identity outside B) and moves x to y.
Bessaga and Pelczyński published a paper in 1969 in which
they prove that a Polish SLH space is CDH. This paper was
submitted for publication in February, 1969.
De Groot published the same result in a paper dated October,
1969.
Topological homogeneity
Countable dense homogeneity
Definition
A space X is called strongly locally homogeneous (abbreviated
SLH) if it has a base B such that for all B ∈ B and x, y ∈ B
there is a homeomorphism f : X → X that is supported on B
(that is, f is the identity outside B) and moves x to y.
Bessaga and Pelczyński published a paper in 1969 in which
they prove that a Polish SLH space is CDH. This paper was
submitted for publication in February, 1969.
De Groot published the same result in a paper dated October,
1969.
Bennett proved in 1972 that every locally compact SLH-space
is CDH.
Topological homogeneity
Countable dense homogeneity
So all of the CDH-spaces that we get from this result are
Polish.
Topological homogeneity
Countable dense homogeneity
So all of the CDH-spaces that we get from this result are
Polish.
This is not by accident. It was shown by Hrušák and Zamora
Avilés in 2005 that Borel spaces that are CDH are Polish.
Under MA+¬CH+ω1 = ω1L there is an analytic CDH-space
that is not complete.
Topological homogeneity
Countable dense homogeneity
So all of the CDH-spaces that we get from this result are
Polish.
This is not by accident. It was shown by Hrušák and Zamora
Avilés in 2005 that Borel spaces that are CDH are Polish.
Under MA+¬CH+ω1 = ω1L there is an analytic CDH-space
that is not complete.
The topological sum of the 1-sphere S1 and S2 is an example
of a CDH-space that is not homogeneous.
Topological homogeneity
Countable dense homogeneity
So all of the CDH-spaces that we get from this result are
Polish.
This is not by accident. It was shown by Hrušák and Zamora
Avilés in 2005 that Borel spaces that are CDH are Polish.
Under MA+¬CH+ω1 = ω1L there is an analytic CDH-space
that is not complete.
The topological sum of the 1-sphere S1 and S2 is an example
of a CDH-space that is not homogeneous.
Bennett proved in 1972 that a connected CDH-space is
homogeneous. (The converse is not true.)
Topological homogeneity
Countable dense homogeneity
So all of the CDH-spaces that we get from this result are
Polish.
This is not by accident. It was shown by Hrušák and Zamora
Avilés in 2005 that Borel spaces that are CDH are Polish.
Under MA+¬CH+ω1 = ω1L there is an analytic CDH-space
that is not complete.
The topological sum of the 1-sphere S1 and S2 is an example
of a CDH-space that is not homogeneous.
Bennett proved in 1972 that a connected CDH-space is
homogeneous. (The converse is not true.)
Hence for connected spaces, countable dense homogeneity can
be thought of as a strong form of homogeneity.
Topological homogeneity
Countable dense homogeneity
The interest in CDH-spaces was kept alive mainly by
Fitzpatrick after 1972.
Topological homogeneity
Countable dense homogeneity
Not all known CDH-spaces are obtained from the Bessaga and
Pelczyński Theorem.
Topological homogeneity
Countable dense homogeneity
Not all known CDH-spaces are obtained from the Bessaga and
Pelczyński Theorem.
Farah, Hrušák and Martı́nez Ranero proved in 2005 that there
is a subspace of R of size ℵ1 that is CDH.
Topological homogeneity
Countable dense homogeneity
Not all known CDH-spaces are obtained from the Bessaga and
Pelczyński Theorem.
Farah, Hrušák and Martı́nez Ranero proved in 2005 that there
is a subspace of R of size ℵ1 that is CDH.
Kawamura, Oversteegen and Tymchatyn proved that complete
Erdős space is CDH. (The complete Erdős space is the set of
all vectors x = (xn )n in Hilbert space 2 such that xn is
irrational for every n.)
Topological homogeneity
Countable dense homogeneity
Not all known CDH-spaces are obtained from the Bessaga and
Pelczyński Theorem.
Farah, Hrušák and Martı́nez Ranero proved in 2005 that there
is a subspace of R of size ℵ1 that is CDH.
Kawamura, Oversteegen and Tymchatyn proved that complete
Erdős space is CDH. (The complete Erdős space is the set of
all vectors x = (xn )n in Hilbert space 2 such that xn is
irrational for every n.)
There is a connected and locally connected Polish CDH-space
which is not SLH (van Mill).
Topological homogeneity
Countable dense homogeneity
Not all known CDH-spaces are obtained from the Bessaga and
Pelczyński Theorem.
Farah, Hrušák and Martı́nez Ranero proved in 2005 that there
is a subspace of R of size ℵ1 that is CDH.
Kawamura, Oversteegen and Tymchatyn proved that complete
Erdős space is CDH. (The complete Erdős space is the set of
all vectors x = (xn )n in Hilbert space 2 such that xn is
irrational for every n.)
There is a connected and locally connected Polish CDH-space
which is not SLH (van Mill).
There is a connected and locally connected Polish CDH-space
S with a dense open rigid connected subset (van Mill). (A
space is rigid if the identity is its only homeomorphism.) In
fact, S × S ≈ 2 .
Topological homogeneity
Countable dense homogeneity
Ungar’s Theorems
Ungar’s Theorems
Topological homogeneity
Countable dense homogeneity
Ungar’s Theorems
Ungar’s Theorems
Ungar published two fundamental papers on homogeneity in
1975 and 1978:
On all kinds of homogeneous spaces, Trans. Amer. Math.
Soc. 212 (1975), 393-400.
Countable dense homogeneity and n-homogeneity, Fund.
Math. 99 (1978), 155-160.
Topological homogeneity
Countable dense homogeneity
Ungar’s Theorems
Ungar’s Theorems
Ungar published two fundamental papers on homogeneity in
1975 and 1978:
On all kinds of homogeneous spaces, Trans. Amer. Math.
Soc. 212 (1975), 393-400.
Countable dense homogeneity and n-homogeneity, Fund.
Math. 99 (1978), 155-160.
Definition
1
2
A space X is n-homogeneous provided that for all subsets F and G
of X of size n there is a homeomorphism f of X such that
f (F ) = G.
A space X is strongly n-homogeneous provided that for all n-tuples
(x1 , . . . , xn ) and (y1 , . . . , yn ) of distinct points of X there is a
homeomorphism f of X such that f (xi ) = yi for all i ≤ n.
Topological homogeneity
Countable dense homogeneity
Ungar’s Theorems
Ungar proved:
Theorem
Let X be a locally compact separable metrizable space such that
no finite set separates X. Then the following statements are
equivalent:
(a) X is CDH.
(b) X is n-homogeneous for every n.
(c) X is strongly n-homogeneous for every n.
Topological homogeneity
Countable dense homogeneity
Ungar’s Theorems
Ungar proved:
Theorem
Let X be a locally compact separable metrizable space such that
no finite set separates X. Then the following statements are
equivalent:
(a) X is CDH.
(b) X is n-homogeneous for every n.
(c) X is strongly n-homogeneous for every n.
Ungar’s basic tool was the Effros Theorem on transitive
actions of Polish groups on Polish spaces. (Every locally
compact homogeneous space admits such an action.)
Topological homogeneity
Countable dense homogeneity
Ungar’s Theorems
Ungar proved:
Theorem
Let X be a locally compact separable metrizable space such that
no finite set separates X. Then the following statements are
equivalent:
(a) X is CDH.
(b) X is n-homogeneous for every n.
(c) X is strongly n-homogeneous for every n.
Ungar’s basic tool was the Effros Theorem on transitive
actions of Polish groups on Polish spaces. (Every locally
compact homogeneous space admits such an action.)
We investigated whether Ungar’s elegant theorem is optimal.
Topological homogeneity
Countable dense homogeneity
Ungar’s Theorems
Ungar proved:
Theorem
Let X be a locally compact separable metrizable space such that
no finite set separates X. Then the following statements are
equivalent:
(a) X is CDH.
(b) X is n-homogeneous for every n.
(c) X is strongly n-homogeneous for every n.
Ungar’s basic tool was the Effros Theorem on transitive
actions of Polish groups on Polish spaces. (Every locally
compact homogeneous space admits such an action.)
We investigated whether Ungar’s elegant theorem is optimal.
The question whether one can prove a similar result with the
assumption of local compactness relaxed to that of
completeness is a natural one in this context.
Topological homogeneity
Countable dense homogeneity
Ungar’s Theorems
Definition
If X is a space, then H (X) denotes the group of
homeomorphisms of X.
Topological homogeneity
Countable dense homogeneity
Ungar’s Theorems
Definition
If X is a space, then H (X) denotes the group of
homeomorphisms of X.
If G ⊆ H (X), then we say that G makes X CDH if for all
countable dense subsets D, E ⊆ X there is an element g ∈ G
such that g(D) = E.
Topological homogeneity
Countable dense homogeneity
Ungar’s Theorems
Definition
If X is a space, then H (X) denotes the group of
homeomorphisms of X.
If G ⊆ H (X), then we say that G makes X CDH if for all
countable dense subsets D, E ⊆ X there is an element g ∈ G
such that g(D) = E.
Similarly for n-homogenous, strongly n-homogeneous, etc.
Topological homogeneity
Countable dense homogeneity
Ungar’s Theorems
Definition
If X is a space, then H (X) denotes the group of
homeomorphisms of X.
If G ⊆ H (X), then we say that G makes X CDH if for all
countable dense subsets D, E ⊆ X there is an element g ∈ G
such that g(D) = E.
Similarly for n-homogenous, strongly n-homogeneous, etc.
Theorem
If the group G makes the space X CDH and no set of size n−1
separates X, then G makes X strongly n-homogeneous.
Topological homogeneity
Countable dense homogeneity
Actions of groups
Example (vM 2008)
There is a homogeneous Polish space X on which no ℵ0 -bounded
topological group acts transitively.
Topological homogeneity
Countable dense homogeneity
Actions of groups
Example (vM 2008)
There is a homogeneous Polish space X on which no ℵ0 -bounded
topological group acts transitively.
A (not necessarily metrizable) topological group G is called
ℵ0 -bounded provided that for every neighborhood U of the
identity e there is a countable subset F of G such that
G = FU.
Topological homogeneity
Countable dense homogeneity
Actions of groups
Example (vM 2008)
There is a homogeneous Polish space X on which no ℵ0 -bounded
topological group acts transitively.
A (not necessarily metrizable) topological group G is called
ℵ0 -bounded provided that for every neighborhood U of the
identity e there is a countable subset F of G such that
G = FU.
It was proved by Guran that a topological group G is
ℵ0 -bounded if and only if it is topologically isomorphic to a
subgroup of a product of separable metrizable groups.
Topological homogeneity
Countable dense homogeneity
Actions of groups
Example (vM 2008)
There is a homogeneous Polish space X on which no ℵ0 -bounded
topological group acts transitively.
A (not necessarily metrizable) topological group G is called
ℵ0 -bounded provided that for every neighborhood U of the
identity e there is a countable subset F of G such that
G = FU.
It was proved by Guran that a topological group G is
ℵ0 -bounded if and only if it is topologically isomorphic to a
subgroup of a product of separable metrizable groups.
X is a tricky subspace of the product of {0, 1}∞ × (0, 1). The
key fact of the example is that its components are wildly
distributed. So the pathology of X is based upon connectivity.
Topological homogeneity
Countable dense homogeneity
Actions of groups
Example (vM 2008)
There is a homogeneous Polish space X on which no ℵ0 -bounded
topological group acts transitively.
A (not necessarily metrizable) topological group G is called
ℵ0 -bounded provided that for every neighborhood U of the
identity e there is a countable subset F of G such that
G = FU.
It was proved by Guran that a topological group G is
ℵ0 -bounded if and only if it is topologically isomorphic to a
subgroup of a product of separable metrizable groups.
X is a tricky subspace of the product of {0, 1}∞ × (0, 1). The
key fact of the example is that its components are wildly
distributed. So the pathology of X is based upon connectivity.
X is 1-homogeneous, but not 2-homogeneous.
Topological homogeneity
Countable dense homogeneity
Actions of groups
Example (vM 2008)
There is a homogeneous Polish space X on which no ℵ0 -bounded
topological group acts transitively.
A (not necessarily metrizable) topological group G is called
ℵ0 -bounded provided that for every neighborhood U of the
identity e there is a countable subset F of G such that
G = FU.
It was proved by Guran that a topological group G is
ℵ0 -bounded if and only if it is topologically isomorphic to a
subgroup of a product of separable metrizable groups.
X is a tricky subspace of the product of {0, 1}∞ × (0, 1). The
key fact of the example is that its components are wildly
distributed. So the pathology of X is based upon connectivity.
X is 1-homogeneous, but not 2-homogeneous.
What does this advertising of X have to do with countable
dense homogeneity?
Topological homogeneity
Countable dense homogeneity
The counterexample
The counterexample
Example
There are a Polish space X and a (separable metrizable)
topological group (G, τ ) such that
1
2
(G, τ ) acts on X by a continuous action, and makes X strongly
n-homogeneous for every n,
X is not CDH.
Topological homogeneity
Countable dense homogeneity
The counterexample
The counterexample
Example
There are a Polish space X and a (separable metrizable)
topological group (G, τ ) such that
1
2
(G, τ ) acts on X by a continuous action, and makes X strongly
n-homogeneous for every n,
X is not CDH.
Hence X ≈ X by the transitive action.
Topological homogeneity
Countable dense homogeneity
The counterexample
The counterexample
Example
There are a Polish space X and a (separable metrizable)
topological group (G, τ ) such that
1
2
(G, τ ) acts on X by a continuous action, and makes X strongly
n-homogeneous for every n,
X is not CDH.
Hence X ≈ X by the transitive action.
The group (G, τ ) cannot be chosen to be complete.
Topological homogeneity
Countable dense homogeneity
The counterexample
The counterexample
Example
There are a Polish space X and a (separable metrizable)
topological group (G, τ ) such that
1
2
(G, τ ) acts on X by a continuous action, and makes X strongly
n-homogeneous for every n,
X is not CDH.
Hence X ≈ X by the transitive action.
The group (G, τ ) cannot be chosen to be complete.
X totally disconnected, i.e, any two distinct points have
disjoint clopen neighborhoods. Moreover, dim X = 1.
Topological homogeneity
Countable dense homogeneity
The counterexample
The counterexample
Example
There are a Polish space X and a (separable metrizable)
topological group (G, τ ) such that
1
2
(G, τ ) acts on X by a continuous action, and makes X strongly
n-homogeneous for every n,
X is not CDH.
Hence X ≈ X by the transitive action.
The group (G, τ ) cannot be chosen to be complete.
X totally disconnected, i.e, any two distinct points have
disjoint clopen neighborhoods. Moreover, dim X = 1.
X is a variation of the example we discussed earlier, the tricky
subspace of the product {0, 1}∞ × (0, 1).
Topological homogeneity
Countable dense homogeneity
The counterexample
The counterexample
Example
There are a Polish space X and a (separable metrizable)
topological group (G, τ ) such that
1
2
(G, τ ) acts on X by a continuous action, and makes X strongly
n-homogeneous for every n,
X is not CDH.
Hence X ≈ X by the transitive action.
The group (G, τ ) cannot be chosen to be complete.
X totally disconnected, i.e, any two distinct points have
disjoint clopen neighborhoods. Moreover, dim X = 1.
X is a variation of the example we discussed earlier, the tricky
subspace of the product {0, 1}∞ × (0, 1).
But there is a significant difference. The components of X are
points, so they are not wildly distributed. The pathology must
be different.
Topological homogeneity
Countable dense homogeneity
The counterexample
X is a tricky subspace of the product {0, 1}∞ × Ec , where Ec
is the complete Erdős space.
Topological homogeneity
Countable dense homogeneity
The counterexample
X is a tricky subspace of the product {0, 1}∞ × Ec , where Ec
is the complete Erdős space.
In 1940 Erdős proved that the ‘rational Hilbert space’ space
E, which consists of all vectors in the real Hilbert space 2
that have only rational coordinates, has dimension one, is
totally disconnected, and is homeomorphic to its own square.
This answered a question of Hurewicz who proved that for
every compact space X and every 1-dimensional space Y we
have that dim(X × Y ) = dim X + 1.
Topological homogeneity
Countable dense homogeneity
The counterexample
X is a tricky subspace of the product {0, 1}∞ × Ec , where Ec
is the complete Erdős space.
In 1940 Erdős proved that the ‘rational Hilbert space’ space
E, which consists of all vectors in the real Hilbert space 2
that have only rational coordinates, has dimension one, is
totally disconnected, and is homeomorphic to its own square.
This answered a question of Hurewicz who proved that for
every compact space X and every 1-dimensional space Y we
have that dim(X × Y ) = dim X + 1.
It is not difficult to prove that E has dimension at most 1.
Erdős proved the surprising fact that every nonempty clopen
subset of E is unbounded, and hence that for no x ∈ E and no
t > 0 the open ball {y ∈ E : x − y
< t} contains a
nonempty clopen subset of E. This implies among other
things that E is nowhere zero-dimensional.
Topological homogeneity
Countable dense homogeneity
The counterexample
X is a tricky subspace of the product {0, 1}∞ × Ec , where Ec
is the complete Erdős space.
In 1940 Erdős proved that the ‘rational Hilbert space’ space
E, which consists of all vectors in the real Hilbert space 2
that have only rational coordinates, has dimension one, is
totally disconnected, and is homeomorphic to its own square.
This answered a question of Hurewicz who proved that for
every compact space X and every 1-dimensional space Y we
have that dim(X × Y ) = dim X + 1.
It is not difficult to prove that E has dimension at most 1.
Erdős proved the surprising fact that every nonempty clopen
subset of E is unbounded, and hence that for no x ∈ E and no
t > 0 the open ball {y ∈ E : x − y
< t} contains a
nonempty clopen subset of E. This implies among other
things that E is nowhere zero-dimensional.
This is the crucial property that makes the Erdős spaces so
interesting.
Topological homogeneity
Countable dense homogeneity
The counterexample
Erdős also proved that the closed subspace Ec of 2 consisting
of all vectors such that every coordinate is in the convergent
sequence {0} ∪ { n1 : n ∈ N} has the same property. The space
Ec is called complete Erdős space and was shown by Dijkstra
to be homeomorphic to the ‘irrational’ Hilbert space, which
consists of all vectors in the real Hilbert space 2 that have
only irrational coordinates. All nonempty clopen subsets of Ec
are unbounded just as the nonempty clopen subsets of E are.
Topological homogeneity
Countable dense homogeneity
The counterexample
Erdős also proved that the closed subspace Ec of 2 consisting
of all vectors such that every coordinate is in the convergent
sequence {0} ∪ { n1 : n ∈ N} has the same property. The space
Ec is called complete Erdős space and was shown by Dijkstra
to be homeomorphic to the ‘irrational’ Hilbert space, which
consists of all vectors in the real Hilbert space 2 that have
only irrational coordinates. All nonempty clopen subsets of Ec
are unbounded just as the nonempty clopen subsets of E are.
The space X is a variation of my previous example X, where
the role of the interval (0, 1) is taken over by the complete
Erdős space.
Topological homogeneity
Countable dense homogeneity
The counterexample
The space Ec surfaces at many places. For example, as the
set of endpoints of certain dendroids (among them, the Lelek
fan), the set of endpoints of the Julia set of the exponential
map, the set of endpoints of the separable universal R-tree,
line-free groups in Banach spaces and Polishable ideals on N.
Topological homogeneity
Countable dense homogeneity
The counterexample
The space Ec surfaces at many places. For example, as the
set of endpoints of certain dendroids (among them, the Lelek
fan), the set of endpoints of the Julia set of the exponential
map, the set of endpoints of the separable universal R-tree,
line-free groups in Banach spaces and Polishable ideals on N.
Metric and topological characterizations of Ec were proved by
Kawamura, Tymchatyn, Oversteegen, Dijkstra and van Mill.
Topological homogeneity
Countable dense homogeneity
The counterexample
The space Ec surfaces at many places. For example, as the
set of endpoints of certain dendroids (among them, the Lelek
fan), the set of endpoints of the Julia set of the exponential
map, the set of endpoints of the separable universal R-tree,
line-free groups in Banach spaces and Polishable ideals on N.
Metric and topological characterizations of Ec were proved by
Kawamura, Tymchatyn, Oversteegen, Dijkstra and van Mill.
Ec is a Polish group, and is CDH, as was shown by
Kawamura, Tymchatyn and Oversteegen.
Topological homogeneity
Countable dense homogeneity
The counterexample
The space Ec surfaces at many places. For example, as the
set of endpoints of certain dendroids (among them, the Lelek
fan), the set of endpoints of the Julia set of the exponential
map, the set of endpoints of the separable universal R-tree,
line-free groups in Banach spaces and Polishable ideals on N.
Metric and topological characterizations of Ec were proved by
Kawamura, Tymchatyn, Oversteegen, Dijkstra and van Mill.
Ec is a Polish group, and is CDH, as was shown by
Kawamura, Tymchatyn and Oversteegen.
Hence Ec = X , but Ec is a building block for X .
Topological homogeneity
Countable dense homogeneity
The counterexample
The space Ec surfaces at many places. For example, as the
set of endpoints of certain dendroids (among them, the Lelek
fan), the set of endpoints of the Julia set of the exponential
map, the set of endpoints of the separable universal R-tree,
line-free groups in Banach spaces and Polishable ideals on N.
Metric and topological characterizations of Ec were proved by
Kawamura, Tymchatyn, Oversteegen, Dijkstra and van Mill.
Ec is a Polish group, and is CDH, as was shown by
Kawamura, Tymchatyn and Oversteegen.
Hence Ec = X , but Ec is a building block for X .
Ec has the following property: every bounded closed subspace
is somewhere zero-dimensional. (This property implies that
Ec ≈ E∞
c .) X contains arbitrarily small closed copies of Ec ,
which are all nowhere zero-dimensional.
Topological homogeneity
Countable dense homogeneity
The counterexample
The space Ec surfaces at many places. For example, as the
set of endpoints of certain dendroids (among them, the Lelek
fan), the set of endpoints of the Julia set of the exponential
map, the set of endpoints of the separable universal R-tree,
line-free groups in Banach spaces and Polishable ideals on N.
Metric and topological characterizations of Ec were proved by
Kawamura, Tymchatyn, Oversteegen, Dijkstra and van Mill.
Ec is a Polish group, and is CDH, as was shown by
Kawamura, Tymchatyn and Oversteegen.
Hence Ec = X , but Ec is a building block for X .
Ec has the following property: every bounded closed subspace
is somewhere zero-dimensional. (This property implies that
Ec ≈ E∞
c .) X contains arbitrarily small closed copies of Ec ,
which are all nowhere zero-dimensional.
This last property of X and a variation of Erős’ original
argument from 1940, give us that X is not CDH.
Topological homogeneity
Countable dense homogeneity
The counterexample
So the pathology of X is not based upon connectivity, but
upon the pathology present in the complete Erdős space.
Topological homogeneity
Open problems
Is there an infinite-dimensional metrizable homogeneous
indecomposable continuum? This is a problem of Rogers from
1985. There is a non-metrizable example, van Mill (1990).
Topological homogeneity
Open problems
Is there an infinite-dimensional metrizable homogeneous
indecomposable continuum? This is a problem of Rogers from
1985. There is a non-metrizable example, van Mill (1990).
If X is a nontrivial homogeneous continuum, is H(X) of
dimension greater than 0? Maybe the pseudo-arc is a
counterexample. It is easy to see that the homeomorphism
groups of solenoids are infinite-dimensional. The
homeomorphism group of a nontrivial manifold without
boundary (both finite and infinite-dimensional) is
infinite-dimensional (also easy), and the homeomorphism
groups of the universal Menger continua are 1-dimensional
(Oversteegen, Tymchatyn, Dijkstra).
Topological homogeneity
Open problems
Is there an infinite-dimensional metrizable homogeneous
indecomposable continuum? This is a problem of Rogers from
1985. There is a non-metrizable example, van Mill (1990).
If X is a nontrivial homogeneous continuum, is H(X) of
dimension greater than 0? Maybe the pseudo-arc is a
counterexample. It is easy to see that the homeomorphism
groups of solenoids are infinite-dimensional. The
homeomorphism group of a nontrivial manifold without
boundary (both finite and infinite-dimensional) is
infinite-dimensional (also easy), and the homeomorphism
groups of the universal Menger continua are 1-dimensional
(Oversteegen, Tymchatyn, Dijkstra).
Is there a Polish uniquely homogeneous space containing more
than one point?
Topological homogeneity
Open problems
Is there an infinite-dimensional metrizable homogeneous
indecomposable continuum? This is a problem of Rogers from
1985. There is a non-metrizable example, van Mill (1990).
If X is a nontrivial homogeneous continuum, is H(X) of
dimension greater than 0? Maybe the pseudo-arc is a
counterexample. It is easy to see that the homeomorphism
groups of solenoids are infinite-dimensional. The
homeomorphism group of a nontrivial manifold without
boundary (both finite and infinite-dimensional) is
infinite-dimensional (also easy), and the homeomorphism
groups of the universal Menger continua are 1-dimensional
(Oversteegen, Tymchatyn, Dijkstra).
Is there a Polish uniquely homogeneous space containing more
than one point?
Is there a nontrivial compact uniquely homogeneous space?
Topological homogeneity
Open problems
Can homogeneous compacta be reconstructed from their
groups of homeomorphisms? That is, are homogeneous
compacta X and Y homeomorphic if and only if H(X) and
H(Y ) are algebraically isomorphic? Ben Ami proved in 2010 a
very interesting partial result. Let G be an abstract group
that acts by homeomorphisms on a space. The action of G is
called locally moving on X if for every nonempty open U in X
there exists an element g ∈ G such that g(x) = x for all
x ∈ U , but g(y) = y for all y ∈ U .
Theorem (Ben Ami, 2010)
Let X and Y be Polish spaces and let G be a group that acts on
X and Y by locally moving actions. If the orbits of the action of G
on X are of the second category in X, and the orbits of the action
of G on Y are of the second category in Y , then X and Y are
homeomorphic.