# Article

Full entry | PDF   (0.2 MB)
Keywords:
remainders in compactifications; homogeneous spaces
Summary:
We prove a dichotomy theorem for remainders in compactifications of homogeneous spaces: given a homogeneous space $X$, every remainder of $X$ is either realcompact and meager or Baire. In addition we show that two other recent dichotomy theorems for remainders of topological groups due to Arhangel'skii cannot be extended to homogeneous spaces.
References:
[1] Arhangel'skii A.V.: Bicompact sets and the topology of spaces. Dokl. Akad. Nauk SSSR 150 (1963), 9--12; MR MR0150733 (27 \#720). MR 0150733
[2] Arhangel'skii A.V.: Two types of remainders of topological groups. Comment. Math. Univ. Carolin. 49 (2008), no. 1, 119--126. MR 2433629
[3] Arhangel'skii A.V.: The Baire property in remainders of topological groups and other results. Comment. Math. Univ. Carolin. 50 (2009), no. 2, 273--279. MR 2537836
[4] Basile D., van Mill J.: A homogeneous space of point-countable type but not of countable type. Comment. Math. Univ. Carolin. 48 (2007), 459--463. MR 2374127
[5] Basile D., van Mill J., Ridderbos G.J.: Sum theorems for Ohio completness. Colloq. Math. 113 (2008), 91--104. DOI 10.4064/cm113-1-6 | MR 2399666
[6] Dow A., Pearl E.: Homogeneity in powers of zero-dimensional first-countable spaces. Proc. Amer. Math. Soc. 125 (1997), no. 8, 2503--2510. DOI 10.1090/S0002-9939-97-03998-1 | MR 1416083 | Zbl 0963.54002
[7] Engelking R.: General Topology. second ed., Heldermann, Berlin, 1989; MR MR2259500. MR 1039321 | Zbl 0684.54001
[8] Gerlits J., Juhász I., Szentmiklóssy Z.: Two improvements on Tkačenko's addition theorem. Comment. Math. Univ. Carolin. 46 (2005), no. 4, 705--710; MR MR2259500 (2008f:54035). MR 2259500 | Zbl 1121.54041
[9] Henriksen M., Isbell J.R.: Some properties of compactifications. Duke Math. J. 25 (1957), 83--105. DOI 10.1215/S0012-7094-58-02509-2 | MR 0096196 | Zbl 0081.38604
[10] Ismail M.: Cardinal functions of homogeneous spaces and topological groups. Math. Japon. 26 (1981), no. 6, 635--646; MR MR649371 (83g:54003). MR 0649371 | Zbl 0479.54003
[11] Juhász I.: Cardinal functions in topology --- ten years later. second ed., Mathematical Centre Tracts, 123, Mathematisch Centrum, Amsterdam, 1980; MR MR 576927 (82a:54002). MR 0576927
[12] Šapirovskii B.E.: $\pi$-character and $\pi$-weight in bicompacta. Dokl. Akad. Nauk SSSR 223 (1975), no. 4, 799--802; MR MR 04110632 (53 \#14380). MR 0410632

Partner of