A remark on Remainders of homogeneous spaces in some compactifications

Document Type : Research Paper


1 Department of Mathematics‎, ‎Shandong Agricultural University‎, ‎Taian 271018‎, ‎China.

2 School of Mathematics‎, ‎Nanjing Normal University‎, ‎Nanjing 210046‎, ‎China.


‎We prove that a remainder $Y$ of a non-locally compact‎ ‎rectifiable space $X$ is locally a $p$-space if and only if‎ ‎either $X$ is a Lindel\"{o}f $p$-space or $X$ is $\sigma$-compact‎, ‎which improves two results by Arhangel'skii‎. ‎We also show that if a non-locally compact‎ ‎rectifiable space $X$ that is locally paracompact has a remainder $Y$ which has locally a $G_{\delta}$-diagonal‎, ‎then both $X$ and $Y$ are separable and metrizable‎, ‎which improves another‎ ‎Arhangel'skii's result‎. ‎It is proved that if a non-locally compact paratopological group $G$ has a locally developable remainder $Y$‎, ‎then either $G$ and $Y$ are separable and metrizable‎, ‎or $G$ is a $\sigma$-compact space with a countable network‎, ‎which improves a result by Wang-He.


Main Subjects

A. V. Arhangel'skii, Topological Function Spaces, Kluwer, Dordrecht-Boston-London, 1992.
A. V. Arhangel'skii, Remainders in compactifications and generalized metrizability properties, Topology Appl. 150 (2005), no. 1-3, 79--90.
A. V. Arhangel'skii, More on remainders close to metrizable spaces, Topology Appl. 154 (2007), no. 6, 1084--1088.
A. V. Arhangel'skii, Two types of remainders of topological groups, Comment. Math. Univ. Carolin. 49 (2008), no. 1, 119--126.
A. V. Arhangel'skii and M. Tkachenko, Topological Groups and Related Structures, Atlantis Press, Amsterdam-Paris, 2008.
A. V. Arhangel'skii, A study of remainders of topological groups, Fund. Math. 203 (2009), no. 2, 165--178.
A. V. Arhangel'skii and M. M. Choban, Remainders of rectifiable spaces, Topology Appl. 157 (2010), no. 4, 789--799.
A. V. Arhangel'skii, Remainders of metrizable spaces and a generalization of Lindelof ∑-spaces, Fund. Math. 157 (2011), no. 1, 87--100.
J. Chaber, Conditions which imply compactness in countably compact spaces, Bull. Pol. Acad. Sci. Math. 24 (1976), no. 11, 993--998.
R. Engelking, General Topology, Revised and Completed Edition, Heldermann Verlag, Berlin, 1989.
G. Gruenhage, Generalized metric spaces, in: Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan, (eds.), North-Holland, Amsterdam, 1984.
A. S. Gul'ko, Recti_able spaces, Topology Appl. 68 (1996), no. 2, 107--112.
M. Henriksen and J. R. Isbell, Some properties of compactifications, Duke Math. J. 25 (1958) 83--106.
C. Liu, Remainders in compactifications of topological groups, Topology Appl. 156 (2009), no. 5, 849--854.
C. Liu, Metrizablity of paratopological (semitopological) groups, Topology Appl. 159 (2012) 1415--1420.
B. E. Shapirovskii, On separability and metrizability of spaces with Souslin's condition, Dokl. Math. 13 (1972) 1633--1638.
B. E. Shapirovskii, Canonical sets and character, density and weight in compact spaces, Dokl. Math. 15 (1974) 1282--1287.
H. F. Wang and W. He, Remainders and cardinal invariants, Topology Appl. 164 (2014) 14--23.