Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
topological group; remainder; compactification; metrizable space; weak base
topological group; remainder; compactification; metrizable space; weak base
Summary:
In this note we first give a summary that on property of a remainder of a non-locally compact topological group $G$ in a compactification $bG$ makes the remainder and the topological group $G$ all separable and metrizable. If a non-locally compact topological group $G$ has a compactification $bG$ such that the remainder $bG\setminus G$ of $G$ belongs to $\mathcal {P}$, then $G$ and $bG\setminus G$ are separable and metrizable, where $\mathcal {P}$ is a class of spaces which satisfies the following conditions: (1) if $X\in \mathcal {P}$, then every compact subset of the space $X$ is a $G_\delta $-set of $X$; (2) if $X\in \mathcal {P}$ and $X$ is not locally compact, then $X$ is not locally countably compact; (3) if $X\in \mathcal {P}$ and $X$ is a Lindelöf $p$-space, then $X$ is metrizable. Some known conclusions on topological groups and their remainders can be obtained from this conclusion. As a corollary, we have that if a non-locally compact topological group $G$ has a compactification $bG$ such that compact subsets of $bG\setminus G$ are $G_{\delta }$-sets in a uniform way (i.e., $bG\setminus G$ is CSS), then $G$ and $bG\setminus G$ are separable and metrizable spaces. In the last part of this note, we prove that if a non-locally compact topological group $G$ has a compactification $bG$ such that the remainder $bG\setminus G$ has a point-countable weak base and has a dense subset $D$ such that every point of the set $D$ has countable pseudo-character in the remainder $bG\setminus G$ (or the subspace $D$ has countable $\pi $-character), then $G$ and $bG\setminus G$ are both separable and metrizable.
References:
[1] Arhangel'skii, A. V.: On a class of spaces containing all metric and locally compact spaces. Mat. Sb. 67 (1965), 55-58 Russian English transl. in: Amer. Math. Soc. Transl. (92) (1970), 1-39. MR 0190889
[2] Arhangel'skii, A. V.: Remainders in compactifications and generalized metrizability properties. Topology Appl. 150 (2005), 79-90. DOI 10.1016/j.topol.2004.10.015 | MR 2133669 | Zbl 1075.54012
[3] Arhangel'skii, A. V.: More on remainders close to metrizable spaces. Topology Appl. 154 (2007), 1084-1088. DOI 10.1016/j.topol.2006.10.008 | MR 2298623 | Zbl 1144.54001
[4] Arhangel'skii, A. V.: Fist countability, tightness, and other cardinal invariants in remainders of topological groups. Topology Appl. 154 (2007), 2950-2961. DOI 10.1016/j.topol.2007.05.013 | MR 2355880
[5] Arhangel'skii, A. V.: Two types of remainders of topological groups. Commentat. Math. Univ. Carol. 49 (2008), 119-126. MR 2433629 | Zbl 1212.54086
[6] Arhangel'skii, A. V.: A study of remainders of topological groups. Fundam. Math. 203 (2009), 165-178. DOI 10.4064/fm203-2-3 | MR 2496236 | Zbl 1182.54043
[7] Arhangel'skii, A. V.: Some aspects of topological algebra and remainders of topological groups. Topol. Proc. 33 (2009), 13-28. MR 2471559 | Zbl 1171.54028
[8] Bennett, H., Byerly, R., Lutzer, D.: Compact $G_{\delta} $ sets. Topology Appl. 153 (2006), 2169-2181. DOI 10.1016/j.topol.2005.08.011 | MR 2239079 | Zbl 1101.54034
[9] Chaber, J.: Conditions which imply compactness in countably compact spaces. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 24 (1976), 993-998. MR 0515000
[10] Douwen, E. K. van, Pfeffer, W. F.: Some properties of the Sorgenfrey line and related spaces. Pac. J. Math. 81 (1979), 371-377. DOI 10.2140/pjm.1979.81.371 | MR 0547605
[11] Engelking, R.: General Topology. Rev. and compl. ed. Sigma Series in Pure Mathematics, 6, Heldermann Verlag, Berlin 1989. MR 1039321 | Zbl 0684.54001
[12] Gerlits, J., Juhász, I., Szentmiklóssy, Z.: Two improvements on Tkačenko's addition theorem. Commentat. Math. Univ. Carol. 46 (2005), 705-710. MR 2259500 | Zbl 1121.54041
[13] Gruenhage, G.: Generalized Metric Spaces. K. Kunen and J. E. Vaughan Handbook of set-theoretic topology. North-Holland, Amsterdam (1984), 423-501. MR 0776629 | Zbl 0555.54015
[14] Gruenhage, G., Michael, E., Tanaka, Y.: Spaces determined by point-countable covers. Pac. J. Math. 113 (1984), 303-332. DOI 10.2140/pjm.1984.113.303 | MR 0749538 | Zbl 0561.54016
[15] Henriksen, M., Isbell, J. R.: Some properties of compactifications. Duke Math. J. 25 (1958), 83-105. DOI 10.1215/S0012-7094-58-02509-2 | MR 0096196 | Zbl 0081.38604
[16] Lin, S., Tanaka, Y.: Point-countable $k$-networks, closed maps, and related results. Topology Appl. 59 (1994), 79-86. DOI 10.1016/0166-8641(94)90101-5 | MR 1293119 | Zbl 0817.54025
[17] Liu, Ch.: On weak bases. Topology Appl. 150 (2005), 91-99. DOI 10.1016/j.topol.2004.11.008 | MR 2133670 | Zbl 1081.54026
[18] Liu, Ch.: Remainders in compactifications of topological groups. Topology Appl. 156 (2009), 849-854. DOI 10.1016/j.topol.2008.09.012 | MR 2498916 | Zbl 1162.54007
[19] Liu, Ch., Lin, S.: Generalized metric spaces with algebaric structures. Topology Appl. 157 (2010), 1966-1974. DOI 10.1016/j.topol.2010.04.010 | MR 2646429
[20] Martin, H. W.: Metrizability of $M$-spaces. Can. J. Math. 25 (1973), 840-841. DOI 10.4153/CJM-1973-086-0 | MR 0328875 | Zbl 0247.54031
[21] Peng, L.-X.: The D-property of some Lindelöf spaces and related conclusions. Topology Appl. 154 (2007), 469-475. DOI 10.1016/j.topol.2006.06.003 | MR 2278697 | Zbl 1110.54014
[22] Peng, L.-X.: A note on $D$-spaces and infinite unions. Topology Appl. 154 (2007), 2223-2227. DOI 10.1016/j.topol.2007.01.020 | MR 2328005 | Zbl 1133.54012
[23] Peng, L.-X., Wang, L. X.: On $ CSS$ spaces and related conclusions. Chinese Acta Math. Sci., Ser. A, Chin. Ed. 30 (2010), 358-363. MR 2664833
[24] Roelcke, W., Dierolf, S.: Uniform Structures on Topological Groups and their Questions. McGraw-Hill International Book Company, New York (1981). MR 0644485
[25] Siwiec, F.: On defining a space by a weak base. Pac. J. Math. 52 (1974), 233-245. DOI 10.2140/pjm.1974.52.233 | MR 0350706 | Zbl 0285.54022
[26] Tkačenko, M.: Introduction to topological groups. Topology Appl. 86 (1998), 179-231. DOI 10.1016/S0166-8641(98)00051-0 | MR 1623960 | Zbl 0955.54013
Search
Partner of
