Polish groups and Baire category methods

Melleray, Julien

Confluentes Mathematici, Tome 8 (2016) no. 1, pp. 89-164.

This article is a slightly modified version of the author’s habilitation thesis, presenting his work on topics related to Polish groups, Baire category methods and metric model theory. Nearly all results presented are not new, though some arguments are. Among new results, we show that, for any countably infinite group $\Gamma$, all conjugacy classes in the space of actions of $\Gamma$ on the Urysohn space are meager; and that the group of bounded isometries of the Urysohn space, endowed with the topology of uniform convergence, is path-connected.

Classification : 22A05, 54E52, 22F50, 54H20, 03E15
Mots clés : Polish groups, automorphism groups, Baire category, topological dynamics, minimal homeomorphisms

[1] Ageev, O. N. Conjugacy of a group action to its inverse, Mat. Zametki, Volume 45 (1989) no. 3, p. 3-11, 127

[2] Ageev, O. N. On extensions of typical group actions (2012) (preprint available at http://arxiv.org/abs/1212.2660)

[3] Ahlbrandt, Gisela; Ziegler, Martin Quasi-finitely axiomatizable totally categorical theories, Ann. Pure Appl. Logic, Volume 30 (1986) no. 1, pp. 63-82 Stability in model theory (Trento, 1984) | DOI

[4] Akin, Ethan Good measures on Cantor space, Trans. Amer. Math. Soc., Volume 357 (2005) no. 7, p. 2681-2722 (electronic)

[5] Anderson, R. D. The algebraic simplicity of certain groups of homeomorphisms, Amer. J. Math., Volume 80 (1958), pp. 955-963

[6] Arens, Richard F.; Eells, James Jr. On embedding uniform and topological spaces, Pacific J. Math., Volume 6 (1956), pp. 397-403

[7] Atim, Alexandru G.; Kallman, Robert R. The infinite unitary and related groups are algebraically determined Polish groups, Topology Appl., Volume 159 (2012) no. 12, pp. 2831-2840 | DOI

[8] Banach, Stefan Théorie des operations linéaires, Chelsea Publishing Co., New York, 1955, vii+254 pages

[9] Becker, Howard; Kechris, Alexander S. The descriptive set theory of Polish group actions, London Mathematical Society Lecture Note Series, 232, Cambridge University Press, Cambridge, 1996, xii+136 pages

[10] Ben Yaacov, Itaï The linear isometry group of the Gurarij space is universal, Proc. Amer. Math. Soc., Volume 142 (2014) no. 7, pp. 2459-2467 | DOI

[11] Ben Yaacov, Itaï; Berenstein, Alexander; Henson, C. Ward; Usvyatsov, Alexander Model theory for metric structures, Model theory with applications to algebra and analysis. Vol. 2 (London Math. Soc. Lecture Note Ser.), Volume 350, Cambridge Univ. Press, Cambridge, 2008, pp. 315-427 | DOI

[12] Ben Yaacov, Itaï; Berenstein, Alexander; Melleray, Julien Polish topometric groups, Trans. Amer. Math. Soc., Volume 365 (2013) no. 7, pp. 3877-3897 | DOI

[13] Ben Yaacov, Itaï; Henson, C. W. Generic orbits and type isolation in the Gurarij space (2014) (to appear in Fund. Math.)

[14] Ben Yaacov, Itaï; Melleray, Julien Grey subsets of Polish spaces, J. Symb. Log., Volume 80 (2015) no. 4, pp. 1379-1397 | DOI

[15] Ben Yaacov, Itaï; Melleray, Julien Grey subsets of Polish spaces, J. Symb. Log., Volume 80 (2015) no. 4, pp. 1379-1397 | DOI

[16] Ben Yaacov, Itaï; Usvyatsov, Alexander Continuous first order logic and local stability, Trans. Amer. Math. Soc., Volume 362 (2010) no. 10, pp. 5213-5259 | DOI

[17] Bezuglyi, S.; Medynets, K. Full groups, flip conjugacy, and orbit equivalence of Cantor minimal systems, Colloq. Math., Volume 110 (2008) no. 2, pp. 409-429 | DOI

[18] Bilge, D.; Melleray, J. Elements of finite order in automorphism groups of Homogeneous structures, Contributions to Discrete Mathematics, Volume 2 (2013)

[19] Bowen, Lewis Measure conjugacy invariants for actions of countable sofic groups, J. Amer. Math. Soc., Volume 23 (2010) no. 1, pp. 217-245 | DOI

[20] Cameron, P. J.; Vershik, A. M. Some isometry groups of the Urysohn space, Annals of Pure and Applied Logic, Volume 143 (2006) no. 1-3, pp. 70-78 | DOI

[21] Chacon, R. V.; Schwartzbauer, T. Commuting point transformations, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, Volume 11 (1969), pp. 277-287

[22] Conze, J. P. Entropie d’un groupe abélien de transformations, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, Volume 25 (1972/73), pp. 11-30

[23] Coskey, Samuel; Ellis, Paul; Schneider, Scott The conjugacy problem for the automorphism group of the random graph, Arch. Math. Logic, Volume 50 (2011) no. 1-2, pp. 215-221 | DOI

[24] de la Rue, Thierry; de Sam Lazaro, José Une transformation générique peut être insérée dans un flot, Ann. Inst. H. Poincaré Probab. Statist., Volume 39 (2003) no. 1, pp. 121-134 | DOI

[25] del Junco, Andrés Disjointness of measure-preserving transformations, minimal self-joinings and category, Ergodic theory and dynamical systems, I (College Park, Md., 1979–80) (Progr. Math.), Volume 10, Birkhäuser Boston, Mass., 1981, pp. 81-89

[26] Delhommé, Christian; Laflamme, Claude; Pouzet, Maurice; Sauer, Norbert Divisibility of countable metric spaces, European J. Combin., Volume 28 (2007) no. 6, pp. 1746-1769 | DOI

[27] Dhal, Heidi Cantor minimal systems and AF-equivalence relations, Norwegian University of Science and Technology (2008) (Ph. D. Thesis available at http://brage.bibsys.no/xmlui/bitstream/handle/11250/249731/123457_FULLTEXT01.pdf?sequence=1)

[28] Diestel, Joseph Sequences and series in Banach spaces, Graduate Texts in Mathematics, 92, Springer-Verlag, New York, 1984, xii+261 pages

[29] Dobrowolski, T.; Toruńczyk, H. Separable complete ANRs admitting a group structure are Hilbert manifolds, Topology Appl., Volume 12 (1981) no. 3, pp. 229-235 | DOI

[30] Downarowicz, Tomasz The Choquet simplex of invariant measures for minimal flows, Israel J. Math., Volume 74 (1991) no. 2-3, pp. 241-256 | DOI

[31] Dye, Henry A. On groups of measure preserving transformations. I, Amer. J. Math., Volume 81 (1959), pp. 119-159

[32] Dye, Henry A. On groups of measure preserving transformations. II, Amer. J. Math., Volume 85 (1963), pp. 551-576

[33] Effros, Edward G. Transformation groups and $C^{*}$-algebras, Annals of Mathematics. Second Series, Volume 81 (1965), pp. 38-55

[34] Fathi, A. Le groupe des transformations de $[0,1]$ qui préservent la mesure de Lebesgue est un groupe simple, Israel J. Math., Volume 29 (1978) no. 2-3, pp. 302-308

[35] Fonf, V. P.; Wojtaszczyk, P. Properties of the Holmes space, Topology Appl., Volume 155 (2008) no. 14, pp. 1627-1633 | DOI

[36] Foreman, Matthew; Weiss, Benjamin An anti-classification theorem for ergodic measure preserving transformations, J. Eur. Math. Soc. (JEMS), Volume 6 (2004) no. 3, pp. 277-292

[37] Fraïssé, Roland Sur l’extension aux relations de quelques propriétés des ordres, Ann. Sci. Ecole Norm. Sup. (3), Volume 71 (1954), pp. 363-388

[38] Fréchet, Maurice Les dimensions d’un ensemble abstrait, Math. Ann., Volume 68 (1910) no. 2, pp. 145-168 | DOI

[39] Fréchet, Maurice L’Expression la Plus Generale de la “Distance” Sur Une Droite, Amer. J. Math., Volume 47 (1925) no. 1, pp. 1-10 | DOI

[40] Friedman, Nathaniel; Gabriel, Patrick; King, Jonathan An invariant for rigid rank-$1$ transformations, Ergodic Theory Dynam. Systems, Volume 8 (1988) no. 1, pp. 53-72 | DOI

[41] Gamarnik, D. Minimality of the group $\mathrm{Autohomeom}\left(C\right)$, Serdica, Volume 17 (1991) no. 4, pp. 197-201

[42] Gao, Su Invariant descriptive set theory, Pure and Applied Mathematics (Boca Raton), 293, CRC Press, Boca Raton, FL, 2009, xiv+383 pages

[43] Gao, Su; Kechris, Alexander S. On the classification of Polish metric spaces up to isometry, Mem. Amer. Math. Soc., Volume 161 (2003) no. 766, viii+78 pages

[44] Gaughan, Edward D. Topological group structures of infinite symmetric groups, Proc. Nat. Acad. Sci. U.S.A., Volume 58 (1967), pp. 907-910

[45] Giordano, Thierry; Pestov, Vladimir Some extremely amenable groups related to operator algebras and ergodic theory, J. Inst. Math. Jussieu, Volume 6 (2007) no. 2, pp. 279-315 | DOI

[46] Giordano, Thierry; Putnam, Ian F.; Skau, Christian F. Topological orbit equivalence and $C^{*}$-crossed products, J. Reine Angew. Math., Volume 469 (1995), pp. 51-111

[47] Giordano, Thierry; Putnam, Ian F.; Skau, Christian F. Full groups of Cantor minimal systems, Israel J. Math., Volume 111 (1999), pp. 285-320

[48] Glasner, E.; Tsirelson, B.; Weiss, B. The automorphism group of the Gaussian measure cannot act pointwise, Israel J. Math., Volume 148 (2005), pp. 305-329 (Probability in mathematics)

[49] Glasner, Eli On minimal actions of Polish groups, Topology Appl., Volume 85 (1998) no. 1-3, pp. 119-125 8th Prague Topological Symposium on General Topology and Its Relations to Modern Analysis and Algebra (1996)

[50] Glasner, Eli The group $\mathrm{Aut}\left(\mu \right)$ is Roelcke precompact, Canad. Math. Bull., Volume 55 (2012) no. 2, pp. 297-302 | DOI

[51] Glasner, Eli; Weiss, Benjamin Weak orbit equivalence of Cantor minimal systems, Internat. J. Math., Volume 6 (1995) no. 4, pp. 559-579

[52] Glasner, Eli; Weiss, Benjamin Topological groups with Rokhlin properties, Colloq. Math., Volume 110 (2008) no. 1, pp. 51-80 | DOI

[53] Godefroy, G.; Kalton, N. J. Lipschitz-free Banach spaces, Studia Math., Volume 159 (2003) no. 1, pp. 121-141 (Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday) | DOI

[54] Grigorchuk, R. I.; Medinets, K. S. On the algebraic properties of topological full groups, Mat. Sb., Volume 205 (2014) no. 6, pp. 87-108

[55] Gromov, M.; Milman, V. D. A topological application of the isoperimetric inequality, Amer. J. Math., Volume 105 (1983) no. 4, pp. 843-854

[56] Gurariĭ, V. I. Spaces of universal placement, isotropic spaces and a problem of Mazur on rotations of Banach spaces, Sibirsk. Mat. Ž., Volume 7 (1966), pp. 1002-1013

[57] Halmos, Paul R.; Samelson, H. On monothetic groups, Proc. Nat. Acad. Sci. U. S. A., Volume 28 (1942), pp. 254-258

[58] Herer, Wojchiech; Christensen, Jens Peter Reus On the existence of pathological submeasures and the construction of exotic topological groups, Math. Ann., Volume 213 (1975), pp. 203-210

[59] Hjorth, Greg Classification and orbit equivalence relations, Mathematical Surveys and Monographs, 75, American Mathematical Society, Providence, RI, 2000, xviii+195 pages

[60] Hjorth, Greg An oscillation theorem for groups of isometries, Geom. Funct. Anal., Volume 18 (2008) no. 2, pp. 489-521 | DOI

[61] Hodges, Wilfrid Model theory, Encyclopedia of Mathematics and its Applications, 42, Cambridge University Press, Cambridge, 1993, xiv+772 pages | DOI

[62] Hodges, Wilfrid; Hodkinson, Ian; Lascar, Daniel; Shelah, Saharon The small index property for $\omega$-stable $\omega$-categorical structures and for the random graph, J. London Math. Soc. (2), Volume 48 (1993) no. 2, pp. 204-218 | DOI

[63] Holmes, M. Randall The universal separable metric space of Urysohn and isometric embeddings thereof in Banach spaces, Fund. Math., Volume 140 (1992) no. 3, pp. 199-223

[64] Holmes, M. Randall The Urysohn space embeds in Banach spaces in just one way, Topology Appl., Volume 155 (2008) no. 14, pp. 1479-1482 | DOI

[65] Hušek, Miroslav Urysohn universal space, its development and Hausdorff’s approach, Topology Appl., Volume 155 (2008) no. 14, pp. 1493-1501 | DOI

[66] Ibarlucía, Tomás; Melleray, Julien Full groups of minimal homeomorphisms and Baire category methods, Ergodic Theory Dynam. Systems, Volume 36 (2016) no. 2, pp. 550-573 | DOI

[67] Kaïchouh, Adriane; Le Maître, François Connected Polish groups with ample generics, Bull. Lond. Math. Soc., Volume 47 (2015) no. 6, pp. 996-1009 | DOI

[68] Kallman, Robert R. A uniqueness result for topological groups, Proc. Amer. Math. Soc., Volume 54 (1976), pp. 439-440

[69] Kallman, Robert R. Uniqueness results for groups of measure preserving transformations, Proceedings of the American Mathematical Society, Volume 95 (1985) no. 1, pp. 87-90 | DOI

[70] Kallman, Robert R. Every reasonably sized matrix group is a subgroup of $S_{\infty }$, Fundamenta Mathematicae, Volume 164 (2000) no. 1, pp. 35-40

[71] Kantorovitch, L. On the translocation of masses, C. R. (Doklady) Acad. Sci. URSS (N.S.), Volume 37 (1942), pp. 199-201

[72] Katětov, M. On universal metric spaces, General topology and its relations to modern analysis and algebra, VI (Prague, 1986) (Res. Exp. Math.), Volume 16, Heldermann, Berlin, 1988, pp. 323-330

[73] Kechris, A. S.; Pestov, V. G.; Todorcevic, S. Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups, Geom. Funct. Anal., Volume 15 (2005) no. 1, pp. 106-189

[74] Kechris, Alexander S. Classical descriptive set theory, Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1995, xviii+402 pages

[75] Kechris, Alexander S. Actions of Polish groups and classification problems, Analysis and logic (Mons, 1997) (London Math. Soc. Lecture Note Ser.), Volume 262, Cambridge Univ. Press, Cambridge, 2002, pp. 115-187

[76] Kechris, Alexander S. Global aspects of ergodic group actions, Mathematical Surveys and Monographs, 160, American Mathematical Society, Providence, RI, 2010, xii+237 pages

[77] Kechris, Alexander S.; Rosendal, Christian Turbulence, amalgamation and generic automorphisms of homogeneous structures, Proc. Lond. Math. Soc., Volume 94 (2007) no. 2, pp. 302-350

[78] Kerr, David Sofic measure entropy via finite partitions, Groups Geom. Dyn., Volume 7 (2013) no. 3, pp. 617-632 | DOI

[79] Kerr, David; Li, Hanfeng; Pichot, Mikaël Turbulence, representations, and trace-preserving actions, Proc. Lond. Math. Soc. (3), Volume 100 (2010) no. 2, pp. 459-484 | DOI

[80] King, Jonathan L. F. The commutant is the weak closure of the powers, for rank-$1$ transformations, Ergodic Theory Dynam. Systems, Volume 6 (1986) no. 3, pp. 363-384 | DOI

[81] King, Jonathan L. F. The generic transformation has roots of all orders, Colloq. Math., Volume 84/85 (2000) no. part 2, pp. 521-547 (Dedicated to the memory of Anzelm Iwanik)

[82] Kittrell, John; Tsankov, Todor Topological properties of full groups, Ergodic Theory Dynam. Systems, Volume 30 (2010) no. 2, pp. 525-545 | DOI

[83] Kubiś, Wiesław; Solecki, Sławomir A proof of uniqueness of the Gurariĭ space, Israel J. Math., Volume 195 (2013) no. 1, pp. 449-456 | DOI

[84] Special issue: Workshop on the Urysohn space, Topology and its Applications, 155 (2008) no. 14 (Held at Ben-Gurion University of the Negev, Beer Sheva, May 21–24, 2006)

[85] Lusky, Wolfgang The Gurarij spaces are unique, Arch. Math. (Basel), Volume 27 (1976) no. 6, pp. 627-635

[86] Mackey, George W. Ergodic theory and virtual groups, Math. Ann., Volume 166 (1966), pp. 187-207

[87] Malicki, Maciej An example of a non non-archimedean Polish group with ample generics, Proc. Amer. Math. Soc., Volume 144 (2016) no. 8, pp. 3579-3581 | DOI

[88] Mazur, Stanislaw Über konvexe mengen in linearen normiertend räumen, Studia Math., Volume 4 (1933), pp. 70-84

[89] Medynets, K. Reconstruction of orbits of Cantor systems from full groups, Bull. Lond. Math. Soc., Volume 43 (2011) no. 6, pp. 1104-1110 | DOI

[90] Melleray, J.; Petrov, F. V.; Vershik, A. M. Linearly rigid metric spaces and the embedding problem, Fund. Math., Volume 199 (2008) no. 2, pp. 177-194 | DOI

[91] Melleray, J.; Tsankov, T. Extremely amenable groups via continuous logic (2013) (research notes, available at http://arxiv.org/abs/1404.4590)

[92] Melleray, Julien Some geometric and dynamical properties of the Urysohn space, Topology Appl., Volume 155 (2008) no. 14, pp. 1531-1560 | DOI

[93] Melleray, Julien A note on Hjorth’s oscillation theorem, Journal of Symbolic Logic, Volume 75 (2010) no. 4, pp. 1359-1365

[94] Melleray, Julien Topology of the isometry group of the Urysohn space, Fund. Math., Volume 207 (2010) no. 3, pp. 273-287 | DOI

[95] Melleray, Julien Extensions of generic measure-preserving actions, Ann. Inst. Fourier (Grenoble), Volume 64 (2014) no. 2, pp. 607-623 http://aif.cedram.org.docelec.univ-lyon1.fr/item?id=AIF_2014__64_2_607_0

[96] Melleray, Julien; Nguyen Van Thé, Lionel; Tsankov, Todor Polish groups with metrizable universal minimal flows, Int. Math. Res. Not. IMRN (2016) no. 5, pp. 1285-1307 | DOI

[97] Melleray, Julien; Tsankov, Todor Generic representations of abelian groups and extreme amenability, Israel J. Math., Volume 198 (2013) no. 1, pp. 129-167 | DOI

[98] Mitchell, Theodore Topological semigroups and fixed points, Illinois J. Math., Volume 14 (1970), pp. 630-641

[99] Nadkarni, M. G. Spectral theory of dynamical systems, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1998, x+182 pages | DOI

[100] Odell, Edward; Schlumprecht, Thomas The distortion problem, Acta Math., Volume 173 (1994) no. 2, pp. 259-281 | DOI

[101] Ornstein, Donald S. On the root problem in ergodic theory, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory, Univ. California Press, Berkeley, Calif. (1972), pp. 347-356

[102] Oxtoby, John C. Invariant measures in groups which are not locally compact, Trans. Amer. Math. Soc., Volume 60 (1946), pp. 215-237

[103] Pestov, Vladimir G. On free actions, minimal flows, and a problem by Ellis, Trans. Amer. Math. Soc., Volume 350 (1998) no. 10, pp. 4149-4165

[104] Pestov, Vladimir G. Ramsey-Milman phenomenon, Urysohn metric spaces, and extremely amenable groups, Israel J. Math., Volume 127 (2002), pp. 317-357

[105] Pestov, Vladimir G. Dynamics of infinite-dimensional groups, University Lecture Series, 40, American Mathematical Society, Providence, RI, 2006, viii+192 pages The Ramsey-Dvoretzky-Milman phenomenon, Revised edition of Dynamics of infinite-dimensional groups and Ramsey-type phenomena [Inst. Mat. Pura. Apl. (IMPA), Rio de Janeiro, 2005; MR2164572]

[106] Pestov, Vladimir G.; Uspenskij, Vladimir V. Representations of residually finite groups by isometries of the Urysohn space, J. Ramanujan Math. Soc., Volume 21 (2006) no. 2, pp. 189-203

[107] Pettis, B. J. On continuity and openness of homomorphisms in topological groups, Ann. of Math. (2), Volume 52 (1950), pp. 293-308

[108] Poizat, Bruno Cours de théorie des modèles, Bruno Poizat, Lyon, 1985, vi+584 pages (Une introduction à la logique mathématique contemporaine. [An introduction to contemporary mathematical logic])

[109] Rosendal, Christian On the non-existence of certain group topologies, Fund. Math., Volume 187 (2005) no. 3, pp. 213-228

[110] Rosendal, Christian Automatic continuity of group homomorphisms, Bull. Symbolic Logic, Volume 15 (2009) no. 2, pp. 184-214 | DOI

[111] Rosendal, Christian A topological version of the Bergman property, Forum Math., Volume 21 (2009) no. 2, pp. 299-332 | DOI

[112] Rosendal, Christian; Solecki, Sławomir Automatic continuity of homomorphisms and fixed points on metric compacta, Israel J. Math., Volume 162 (2007), pp. 349-371

[113] Sabok, Marcin Automatic continuity for isometry groups (2013) (preprint available at http://www.math.uni.wroc.pl/~sabok/PRACE/automatic.rev.pdf)

[114] Schoretsanitis, Konstantinos Fraïssé theory for metric structures, University of Illinois at Urbana-Champaign (2007) (Ph. D. Thesis)

[115] Schreier, J.; Ulam, S. Über die Permutationsgruppe der natürlichen Zachlenfolge, Studia Math., Volume 4 (1933), pp. 134-141

[116] Sierpiński, Waclaw Sur un espace métrique séparable universel, Fund. Math., Volume 33 (1945), pp. 115-122

[117] Solecki, Sławomir Closed subgroups generated by generic measure automorphisms, Ergodic Theory Dynam. Systems, Volume 34 (2014) no. 3, pp. 1011-1017 | DOI

[118] Stepin, A. M.; Eremenko, A. M. Nonuniqueness of an inclusion in a flow and the vastness of a centralizer for a generic measure-preserving transformation, Mat. Sb., Volume 195 (2004) no. 12, pp. 95-108 | DOI

[119] Tent, Katrin; Ziegler, Martin A course in model theory, Lecture Notes in Logic, 40, Association for Symbolic Logic, La Jolla, CA, 2012, x+248 pages

[120] Tent, Katrin; Ziegler, Martin The isometry group of the bounded Urysohn space is simple, Bull. Lond. Math. Soc., Volume 45 (2013) no. 5, pp. 1026-1030 | DOI

[121] Tent, Katrin; Ziegler, Martin On the isometry group of the Urysohn space, J. Lond. Math. Soc. (2), Volume 87 (2013) no. 1, pp. 289-303 | DOI

[122] Thomas, Simon Infinite products of finite simple groups. II, J. Group Theory, Volume 2 (1999) no. 4, pp. 401-434 | DOI

[123] Tsankov, Todor Automatic continuity for the unitary group, Proc. Amer. Math. Soc., Volume 141 (2013) no. 10, pp. 3673-3680 | DOI

[124] Ulam, S. M. A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York-London, 1960, xiii+150 pages

[125] Urysohn, P. Sur un espace métrique universel., C. R., Volume 180 (1925), pp. 803-806

[126] Uspenskij, Vladimir On the group of isometries of the Urysohn universal metric space, Comment. Math. Univ. Carolin., Volume 31 (1990) no. 1, pp. 181-182

[127] Uspenskij, Vladimir The Urysohn universal metric space is homeomorphic to a Hilbert space, Topology Appl., Volume 139 (2004) no. 1-3, pp. 145-149

[128] van Mill, Jan Infinite-dimensional topology, North-Holland Mathematical Library, 43, North-Holland Publishing Co., Amsterdam, 1989, xii+401 pages

[129] Weaver, Nik Lipschitz algebras, World Scientific Publishing Co. Inc., River Edge, NJ, 1999, xiv+223 pages

[130] Wei, T.-J Descriptive properties of measure preserving actions and the associated unitary representations, Caltech (2005) (Ph. D. Thesis)

[131] Wesolek, Phillip Conjugacy class conditions in locally compact second countable groups, Proc. Amer. Math. Soc., Volume 144 (2016) no. 1, pp. 399-409 | DOI

[132] Zatitskiĭ, P. B. Canonical embeddings of compact metric spaces, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Volume 378 (2010) no. Teoriya Predstavlenii, Dinamicheskie Sistemy, Kombinatornye Metody. XVIII, p. 40-46, 229 | DOI

[133] Zucker, Andy Topological dynamics of automorphism groups, ultrafilter combinatorics, and the generic point problem, Trans. Amer. Math. Soc., Volume 368 (2016) no. 9, pp. 6715-6740 | DOI