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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.