On sequential analytic groups

Shibakov, Alexander

doi:10.1090/proc/13514

On sequential analytic groups

Author: Alexander Y. Shibakov
Journal: Proc. Amer. Math. Soc. 145 (2017), 4087-4096
MSC (2010): Primary 54D55, 54H05; Secondary 54A20
DOI: https://doi.org/10.1090/proc/13514
Published electronically: March 27, 2017
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Abstract: We answer a question of S. Todorčević and C. Uzcátegui from their 2005 work by showing that the only possible sequential orders of sequential analytic groups are and $\omega _1$ . Other results on the structure of sequential analytic spaces and their relation to other classes of spaces are given as well. In particular, we provide a full topological classification of sequential analytic groups by showing that all such groups are either metrizable or $k_\omega$ -spaces, which, together with a result by Zelenyuk, implies that there are exactly $\omega _1$ non-homeomorphic analytic sequential group topologies.

[1] Taras Banakh and Lubomyr Zdomskyĭ, The topological structure of (homogeneous) spaces and groups with countable 𝑐𝑠*-character, Appl. Gen. Topol. 5 (2004), no. 1, 25–48. MR 2087279, https://doi.org/10.4995/agt.2004.1993
[2] Doyel Barman and Alan Dow, Proper forcing axiom and selective separability, Topology Appl. 159 (2012), no. 3, 806–813. MR 2868880, https://doi.org/10.1016/j.topol.2011.11.048
[3] Eric K. van Douwen, The product of a Fréchet space and a metrizable space, Topology Appl. 47 (1992), no. 3, 163–164. MR 1192305, https://doi.org/10.1016/0166-8641(92)90026-V
[4] Eric K. van Douwen, The integers and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111–167. MR 776622
[5] M. Hrušák and U. A. Ramos-García, Malykhin’s problem, Adv. Math. 262 (2014), 193–212. MR 3228427, https://doi.org/10.1016/j.aim.2014.05.009
[6] V. Kannan, Ordinal invariants in topology, Mem. Amer. Math. Soc. 32 (1981), no. 245, v+164. MR 617500, https://doi.org/10.1090/memo/0245
[7] Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597
[8] Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
[9] E. Michael, ℵ₀-spaces, J. Math. Mech. 15 (1966), 983–1002. MR 0206907
[10] Peter J. Nyikos, Metrizability and the Fréchet-Urysohn property in topological groups, Proc. Amer. Math. Soc. 83 (1981), no. 4, 793–801. MR 630057, https://doi.org/10.1090/S0002-9939-1981-0630057-2
[11] Roberto Peirone, Regular semitopological groups of every countable sequential order, Topology Appl. 58 (1994), no. 2, 145–149. MR 1283584, https://doi.org/10.1016/0166-8641(94)90127-9
[12] I. Protasov and E. Zelenyuk, Topologies on groups determined by sequences, Mathematical Studies Monograph Series, vol. 4, VNTL Publishers, L′viv, 1999. MR 1718814
[13] A. Shibakov, No interesting sequential groups, preprint, arXiv:1604.08868 [math.GN]
[14] Alexander Shibakov, Sequential topological groups of any sequential order under CH, Fund. Math. 155 (1998), no. 1, 79–89. MR 1487989
[15] Alexander Shibakov, Metrizability of sequential topological groups with point-countable 𝑘-networks, Proc. Amer. Math. Soc. 126 (1998), no. 3, 943–947. MR 1443165, https://doi.org/10.1090/S0002-9939-98-04139-2
[16] Stevo Todorčević, Analytic gaps, Fund. Math. 150 (1996), no. 1, 55–66. MR 1387957
[17] S. Todorčević and C. Uzcátegui, Analytic 𝑘-spaces, Topology Appl. 146/147 (2005), 511–526. MR 2107168, https://doi.org/10.1016/j.topol.2003.09.013
[18] Stevo Todorčević and Carlos Uzcátegui, Analytic topologies over countable sets, Topology Appl. 111 (2001), no. 3, 299–326. MR 1814231, https://doi.org/10.1016/S0166-8641(99)00223-0
[19] E. G. Zelenyuk, Topologies on groups, defined by compacta, Mat. Stud. 5 (1995), 5–16, 124 (Russian, with English and Russian summaries). MR 1691086