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Keywords:
locally compact; connected; sequentially connected; Pontryagin duality; torsion-free; divisible; metrizable element; extension of a group
locally compact; connected; sequentially connected; Pontryagin duality; torsion-free; divisible; metrizable element; extension of a group
Summary:
We prove that every connected locally compact Abelian topological group is sequentially connected, i.e., it cannot be the union of two proper disjoint sequentially closed subsets. This fact is then applied to the study of extensions of topological groups. We show, in particular, that if $H$ is a connected locally compact Abelian subgroup of a Hausdorff topological group $G$ and the quotient space $G/H$ is sequentially connected, then so is $G$.
References:
[1] Arhangel'skii A.V., Tkachenko M.G.: Topological Groups and Related Structures. Atlantis Series in Mathematics, Vol. I, Atlantis Press and World Scientific, Paris–Amsterdam, 2008. MR 2433295
[2] Davis H.F.: A note on Haar measure. Proc. Amer. Math. Soc. 6 (1955), 318–321. DOI 10.1090/S0002-9939-1955-0069187-X | MR 0069187 | Zbl 0068.25608
[3] Engelking R.: General Topology. Heldermann Verlag, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[4] Fedeli A., Le Donne A.: On good connected preimages. Topology Appl. 125 (2002), 489–496. DOI 10.1016/S0166-8641(01)00294-2 | MR 1935165 | Zbl 1019.54008
[5] Hewitt E., Ross K.A.: Abstract Harmonic Analysis, Volume I. Springer, Berlin–Göttingen–Heidelberg, 1979. MR 0551496 | Zbl 0837.43002
[6] Huang Q., Lin S.: Notes on sequentially connected spaces. Acta Math. Hungar. 110 (2006), 159–164. DOI 10.1007/s10474-006-0012-1 | MR 2198420 | Zbl 1100.54015
[7] Ivanovskiĭ L.N.: On a hypothesis of P.S. Alexandrov. Dokl. Akad. Nauk SSSR 123 (1958), 785–786 (in Russian). MR 0102569
[8] Kuz'minov V.: On a hypothesis of P.S. Alexandrov in the theory of topological groups. Dokl. Akad. Nauk SSSR 125 (1959), 727–729 (in Russian). MR 0104753
[9] Lin S.: The images of connected metric spaces. Chinese Ann. Math. A 26 (2005), 345–350. MR 2158870 | Zbl 1081.54016
[10] Lin S., Lin F.C., Xie L.H.: The extensions of topological groups about convergence phenomena. preprint.
[11] Pontryagin L.S.: Continuous Groups. third edition, Nauka, Moscow, 1973. MR 0767087 | Zbl 0659.22001
[12] Robinson D.J.F.: A Course in the Theory of Groups. Springer, Berlin, 1982. MR 0648604 | Zbl 0836.20001
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