拓扑空间的剩余与广义度量化性质
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摘要
本文研究拓扑空间(包括拓扑群,半拓扑群,仿拓扑群以及齐性空间)在紧化空间中的剩余问题,主要讨论剩余的广义度量化性质与空间自身的广义度量化性质之间的关系.此外,文章还对K-空间及拓扑群同胚等问题进行了研究.
第1章给出了本文研究的历史背景、主要结果及预备知识.在第1节,我们主要回顾了国内外在紧化剩余方面的研究历史、现状,给出了本文的主要结果.第2节给出了文中用到的一些基本概念及一些相关命题.
第2章研究了一般拓扑空间及仿拓扑群的紧化剩余,改进和推广了A.VArhangel'skii的两个结果,得到了如下结果：
(a)设bG是非局部紧仿拓扑群G的一个紧化空间,则Y=bG\G是局部p-空间当且仅当下列两个条件中的一个成立：(1)G是Lindelof p-空间；(2)G是σ-紧空间.
(b)设bX是无处局部紧的完全正则空间X的一个紧化空间,如果满足：(1)X具有局部Gδ-对角线；(2)Y=bX\X是仿紧p-空间,则X是可分的度量空间.
第3章讨论了半拓扑群与Rectifiable空间的紧化剩余,改进和推广了A.VArhangel'skii及刘川的几个结果,主要结果如下：
(c)设bG是非局部紧半拓扑群G的一个紧化空间,如果Y=bG\G具有局部点可数基,则G与bG都具有可数基.
(d)设bX是非局部紧的局部仿紧rectifiable空间X的一个紧化空间,如果Y=bX\X具有局部Gδ-对角线,则X与Y都具有可数基.
第4章讨论了K-空间及CS-空间,肯定地回答了由V.IMalykhin与G.Tironi在文献[62]中提出的一个公开问题：
(e)紧的K-空间具有可数紧度吗?
此外本章还讨论了K-拓扑群及CS-空间的映射性质.
第5章讨论了拓扑群的同胚问题,解决了A.V.Arhangel'skii与M.Tkachenko在文献[33]中提出的两个公开问题：
(f)每一个连通的拓扑群都是w-narrow拓扑群吗?
(g)设e是零维拓扑群G的单位元,子空间G、｛e｝一定与某个拓扑群同胚吗?
In this thesis, remainders of topological spaces (includes paratopological groups, semitopological groups and homogeneous spaces) in some compactifications are stud-ied. We mainly investigate the relationship between the generalized metrizability properties of remainders and the generalized metrizability properties of the space it-self. In addition, we also discuss K-spaces and the homeomorphisms of topological groups.
In Chapter1, some background, main results and preliminaries are given. In Section1, we recall the research progress of remainders of topological spaces at home and aboard, and offer the main results in this thesis. In Section2, we introduce some preliminary knowledge, including some important concepts and related propositions.
In Chapter2, we investigate remainders of general topological spaces and paratopological groups, two theorems by A.V. Arhangel'skii are improved. The fol-lowing two main results are established:
(a) If bG is a compactification of a non-locally compact paratopological group G, then Y=bG\G is locally a p-space if and only if at least one of the following conditions holds:(1) G is a Lindelof p-space;(2) G is σ-compact.
(b) If bX is a compactification of a nowhere locally compact space X with locally a Gδ-diagonal such that Y=bX\X is a paracompact p-space, then X is separable and metrizable.
In Chapter3, we study remainders of semitopological groups and rectifiable spaces, several results given by A.V. Arhangel'skii and Chuan Liu are improved and generalized. Main results are as follows:
(c) If a non-locally compact semitopological group G has a compatification bG such that Y=bG\G has locally a point-countable base, then both G and bG are separable and metrizable.
(d) If a non-locally compact locally paracompact rectifiable space X has a com-pactification bX satisfying that Y=bX\X has locally a Gδ-diagonal, then X and Y are both have countable bases.
In Chapter4, K-spaces and CS-spaces are discussed. An open problem posed by V.I. Malykhin and G. Tironi in [62] is answered firmly:
(e) Must a compact K-space have countable tightness?
We also investigate K-topological groups and the mapping properties of CS-spaces.
In Chapter5, we investigate the homeomorphisms of topological groups, and two open problems posed by A.V. Arhangel'skii and M. Tkachenko in [33] are solved:
(f) Is every connected topological group homeomorphic to an w-narrow topolog-ical group?
(g) For a zero-dimensional topological group G with neutral element e, must X=G\{e} be homeomorphic to a topological group?
第1章给出了本文研究的历史背景、主要结果及预备知识.在第1节,我们主要回顾了国内外在紧化剩余方面的研究历史、现状,给出了本文的主要结果.第2节给出了文中用到的一些基本概念及一些相关命题.
第2章研究了一般拓扑空间及仿拓扑群的紧化剩余,改进和推广了A.VArhangel'skii的两个结果,得到了如下结果：
(a)设bG是非局部紧仿拓扑群G的一个紧化空间,则Y=bG\G是局部p-空间当且仅当下列两个条件中的一个成立：(1)G是Lindelof p-空间；(2)G是σ-紧空间.
(b)设bX是无处局部紧的完全正则空间X的一个紧化空间,如果满足：(1)X具有局部Gδ-对角线；(2)Y=bX\X是仿紧p-空间,则X是可分的度量空间.
第3章讨论了半拓扑群与Rectifiable空间的紧化剩余,改进和推广了A.VArhangel'skii及刘川的几个结果,主要结果如下：
(c)设bG是非局部紧半拓扑群G的一个紧化空间,如果Y=bG\G具有局部点可数基,则G与bG都具有可数基.
(d)设bX是非局部紧的局部仿紧rectifiable空间X的一个紧化空间,如果Y=bX\X具有局部Gδ-对角线,则X与Y都具有可数基.
第4章讨论了K-空间及CS-空间,肯定地回答了由V.IMalykhin与G.Tironi在文献[62]中提出的一个公开问题：
(e)紧的K-空间具有可数紧度吗?
此外本章还讨论了K-拓扑群及CS-空间的映射性质.
第5章讨论了拓扑群的同胚问题,解决了A.V.Arhangel'skii与M.Tkachenko在文献[33]中提出的两个公开问题：
(f)每一个连通的拓扑群都是w-narrow拓扑群吗?
(g)设e是零维拓扑群G的单位元,子空间G、｛e｝一定与某个拓扑群同胚吗?
In this thesis, remainders of topological spaces (includes paratopological groups, semitopological groups and homogeneous spaces) in some compactifications are stud-ied. We mainly investigate the relationship between the generalized metrizability properties of remainders and the generalized metrizability properties of the space it-self. In addition, we also discuss K-spaces and the homeomorphisms of topological groups.
In Chapter1, some background, main results and preliminaries are given. In Section1, we recall the research progress of remainders of topological spaces at home and aboard, and offer the main results in this thesis. In Section2, we introduce some preliminary knowledge, including some important concepts and related propositions.
In Chapter2, we investigate remainders of general topological spaces and paratopological groups, two theorems by A.V. Arhangel'skii are improved. The fol-lowing two main results are established:
(a) If bG is a compactification of a non-locally compact paratopological group G, then Y=bG\G is locally a p-space if and only if at least one of the following conditions holds:(1) G is a Lindelof p-space;(2) G is σ-compact.
(b) If bX is a compactification of a nowhere locally compact space X with locally a Gδ-diagonal such that Y=bX\X is a paracompact p-space, then X is separable and metrizable.
In Chapter3, we study remainders of semitopological groups and rectifiable spaces, several results given by A.V. Arhangel'skii and Chuan Liu are improved and generalized. Main results are as follows:
(c) If a non-locally compact semitopological group G has a compatification bG such that Y=bG\G has locally a point-countable base, then both G and bG are separable and metrizable.
(d) If a non-locally compact locally paracompact rectifiable space X has a com-pactification bX satisfying that Y=bX\X has locally a Gδ-diagonal, then X and Y are both have countable bases.
In Chapter4, K-spaces and CS-spaces are discussed. An open problem posed by V.I. Malykhin and G. Tironi in [62] is answered firmly:
(e) Must a compact K-space have countable tightness?
We also investigate K-topological groups and the mapping properties of CS-spaces.
In Chapter5, we investigate the homeomorphisms of topological groups, and two open problems posed by A.V. Arhangel'skii and M. Tkachenko in [33] are solved:
(f) Is every connected topological group homeomorphic to an w-narrow topolog-ical group?
(g) For a zero-dimensional topological group G with neutral element e, must X=G\{e} be homeomorphic to a topological group?
引文
[1]B. Alleche, A. V. Arhangel'skii, J. Calbrix, Weak developments and metrization, Topology Appl.100 (2000),23-38.
[2]A.V. Arhangel'skii, On a class of spaces containing all metric and all locally compact spaces, Mat. Sb.67 (1965),55-58 (In Russian); English transl. in: Amer. Math. Soc. Transl.92 (1970),1-39.
[3]A.V. Arhangel'skii, Relations among the invariants of topological groups and their subspaces, Russian Math. Surveys,35 (1980),1-23.
[4]A.V. Arhangel'skii, Classes of topological groups, Russian Math. Surveys 36, 3(1981),151-174.
[5]A.V. Arhangel'skii, Topological Function Spaces, Kluwer,1992.
[6]A.V. Arhangel'skii, Some connections between properties of topological groups and of their remainders. Moscow Univ. Math. Bull.54,3 (1999),1-6.
[7]A.V. Arhangel'skii, Strong disconnectedness properties and remainders in com-pactifications, Topology Appl.107 (2000),3-12.
[8]A.V. Arhangel'skii, Topological invariants in algebraic environment, Recent progress in general topology Ⅲ, by Husek and van Mill,2002,1-58.
[9]A.V. Arhangel'skii, Remainders in compactifications and generalized metriz-ability properties, Topology Appl.150 (2005),79-90.
[10]A.V. Arhangel'skii, Quotients with respect to locally compact subgroups, Hous-ton J. Math.31,1 (2005),215-226.
[11]A.V. Arhangel'skii, More on remainders close to metrizable spaces, Topology Appl.154 (2007),1084-1088.
[12]A.V. Arhangel'skii, First countability, tightness, and other cardinal invariants in remainders of topological groups, Topology Appl.154 (2007),2950-2961.
[13]A.V. Arhangel'skii, Two types of remainders of topological groups, Comment. Math. Univ. Carolin.49,1 (2008),119-126.
[14]A.V. Arhangel'skii, A study of remainders of topological groups, Fund. Math. 105 (2009),1-14.
[15]A.V. Arhangel'skii, Gδ-points in remainders of topological groups and some addition theorems in compacta, Topology Appl.156 (2009),2013-2018.
[16]A.V. Arhangel'skii, Some aspects of topological algebra and remainders of topological groups, Topology Proc.33 (2009),13-28.
[17]A.V. Arhangel'skii, The Baire properties in remainders of topological groups and other results, Comment. Math. Univ. Carolin.50,2 (2009),273-279.
[18]A.V. Arhangel'skii, A construction of a Frechet-Urysohn space, and some con-vergence concepts, Comment. Math. Univ. Carolin.51,1 (2010),99-112.
[19]A.V. Arhangel'skii, Remainders of metrizable spaces and a generalization of Lindelof E-spaces, Fund. Math.215 (2011),87-100.
[20]A.V. Arhangel'skii, Components of first-countability and various kinds of pseu-doopen mappings, Topology Appl.158 (2011),215-222.
[21]A.V. Arhangel'skii, A. Bella, Cardinal invariants in remainders and variations of tightness, Proc. Amer. Math. Soc.119,3 (1993),947-954.
[22]A.V. Arhangel'skii, A. Bella, Countable fan-tightness versus countable tight-ness, Comment. Math. Univ. Carolin.37,3 (1996),565-576.
[23]A.V. Arhangel'skii, A. Bella, On pseudocompactness of remainders of topolog-ical groups and some classes of mappings, Topology Appl. 111 (2001),21-33.
[24]A.V. Arhangel'skii, M.M. Choban, On remainders of rectifiable spaces, Topol-ogy Appl.157 (2010),789-799.
[25]A.V. Arhangel'skii, M.M. Choban, Completeness type properties of semitopo-logical groups, and the theorems of Montgomery and Ellis, Topology Proc.37 (2011),1-28.
[26]A.V. Arhangel'skii, M.M. Choban, E.P. Mihaylova, About homogeneous spaces and conditions of completeness of spaces. Math. and Education in Math. 41 (2012),129-133.
[27]A.V. Arhangel'skii, M.M. Choban, E.P. Mihaylova, About homogeneous spaces and Baire property in remainders, Math. and Education in Math.41 (2012),134-138.
[28]A.V. Arhangel'skii, M.M. Choban, E.P. Mihaylova, About topological groups and the Baire properties in remainders, Proceedings of the Forty First Spring Conference of the Union of Bulgarian Mathematicians Borovetz, April 9-12, 2012.
[29]A.V. Arhangel'skii, M.M. Choban, E.P. Mihaylova, About homogeneous spaces and the Baire properties in remainders, Proceedings of the Forty First Spring Conference of the Union of Bulgarian Mathematicians Borovetz, April 9-12,2012.
[30]A.V. Arhangel'skii, W. Just, E. Rezniczenko, P. Szeptycki, Sharp bases and weakly uniform bases versus point-countable bases, Topology Appl.100 (2000),39-46.
[31]A.V. Arhangel'skii, J.Van Mill, On topological groups with a first-countable remainder, Topology Proc.42 (2013),157-163.
[32]A.V. Arhangel'skii, E.V. Reznichenko, Paratopological and semitopological groups versus topological groups, Topology. Appl.151 (2005),107-119.
[33]A.V. Arhangel'skii, M.Tkachenko, Topological Groups and Related Structures, Atlantis Press, World Sci.,2008.
[34]D. Basile, A. Bella, About remainders in compactifications of homogeneous spaces, Comment. Math. Univ. Carolin.50,4 (2009),607-613.
[35]D. Basile, J.Van Mill, A homogeneous space of point-countable but not of countable type, Comment. Math. Univ. Carolin.48,3 (2007),459-463.
[36]D. Basile, J.Van Mill, Ohio completeness and products, Topology Appl.155 (2008),180-189.
[37]A. Bouziad, Every Cech-analytic Baire semitopological group is a topological group, Proc. Amer. Math. Soc.24,3 (1998),953-959.
[38]J. Calbrix, B. Alleche, Multifunctions and Cech-complete spaces, Topology Atlas (Proceeding of the 8th Prague Topological Symposium),1997,30-36.
[39]J. Chaber, Conditions which imply compactness in countably compact spaces, Bull. Pol. Acad. Math.,24 (1976),993-998.
[40]M.M. Coban, The structure of locally compact algebras, Serdica 18 (1992), 129-137.
[41]R. Ellis, A note on the continuity of the inverse, Proc. Amer. Math. Soc.8 (1957),372-373.
[42]R. Engelking, General Topology, Revised and completed edition, Heldermann Verlag Berlin,1989.
[43]V.V. Filippov, On perfect images of paracompact p-spaces, Soviet Math. Dok1. 176 (1967),533-536.
[44]G. Gruenhage, Generalized metric spaces, in:K. Kunen, J.E. Vaughan (Eds.), Handbook of Set-Theoretic Topology, North-Holland,1984,423-501.
[45]A.S. Gul'ko, Rectifiable spaces, Topology Appl.68 (1996),107-112.
[46]I.I. Guran, On topological groups close to being Lindelof, Soviet Math. Dokl. 23 (1981),173-175.
[47]S. Hartman, J. Mycielski, On embeddings of topological groups into connected topological groups, Colloq. Math.5 (1958),167-169.
[48]R.W. Heath and W.F. Lindgren, Weakly uniform bases, Houston J. Math.2 (1976),85-90.
[49]M. Henriksen, J.R. Isbell, Some properties of compactifications, Duke Math. J. 25 (1958),83-106.
[50]R. Hodel, Cardinal functions I, in:K. Kunen, J.E. Vaughan (Eds.), Handbook of Set-Theoretic Topology, North-Holland,1984,3-61.
[51]I. Juhasz, Z. Szentmiklossy, Convergent free sequences in compact spaces, Proc. Amer. Math. Soc.116,4 (1992),1153-1160.
[52]Y. Kodama, K. Nagami, General Topology, Iwanami Shoten, Tokyo,1974.
[53]N. Lasnev, Closed images of metric spaces, Dokl. Akad. Nauk. SSSR,170 (1966),505-507.
[54]F.C. Lin, Local properties on the remainders of the topological groups, Kodai Math. J.34,3 (2011),505-518.
[55]F.C. Lin, S.X. Lin, About remainders in compactifications of paratopological groups, ArXiv:1106.3836v1 [Math. GN] 20 June 2011.
[56]F.C. Lin, R.X. Shen, On rectifiable spaces and paratopological groups, Topol-ogy Appl.158 (2011),597-610.
[57]P. Li, L. Mou and S. Wang, Notes on questions about spaces with algebraic structures, Topology Appl.159 (2012),3619-3623.
[58]S. Lin, Generalized metric spaces and mappings (lnd edition), China Science Press, Beijing,1995 (in Chinese).
[59]C. Liu, Remainders in compactifications of topological groups, Topology Appl. 156 (2009),849-854.
[60]C. Liu, Metrizablity of paratopological (semitopological) groups, Topology Appl.159 (2012),1415-1420.
[61]C. Liu, S. Lin, Generalized metric spaces with algebraic structures, Topology Appl.157 (2010),1966-1974.
[62]V.I. Malykhin, G. Tironi, Weakly Frechet-Urysohn and Pytkeev spaces, Topol-ogy Appl.104 (2000),181-190.
[63]K. Nagami, Σ-spaces, Fund. Math.65 (1969),169-192.
[64]P.J. Nyikos, Metrizability and the Frechet-Urysohn property in topological groups, Proc. Amer. Math. Soc.83 (1981),793-801.
[65]B.A. Pasynkov, Almost-metrizable topological groups, Dokl. Akad. Nauk SSSR 161 (1965),281-284.
[66]L.X. Peng and Y.F. He, A note on topological groups and their remainders, Czechoslovak Math. J.137,62 (2012),197-214.
[67]V.I. Ponomarev, Axioms of countably and continous mappings, Bull. Polon. Sci. Ser. Math. Astr. Phys.8 (1960),127-133.
[68]V.I. Ponomarev, On metrizability of Lindelof p-spaces with a point-countable base, Dokl. Akad. Nauk SSSR 174 (1967),1274-1277.
[69]L.S. Pontryagin, Continuous Groups (Third edition), Nauka, Moscow,1973.
[70]E.G. Pytkeev, On maximally resolvable spaces, Proc. Steklov Inst. Math.4 (1984),225-230.
[71]E.A. Reznichenko, Extensions of functions defined products of pseudocom-pact spaces and continuity of the converse in pseudocompact groups, Topology Appl.59 (1994),233-244.
[72]W. Roelke, S. Dierolf, Uniform Structures on Topological Groups and Their Quotients, McGraw-Hill, New York,1981.
[73]B.E. Shapirovskij, On separability and metrizability of spaces with Suslin's condition, Soviet Math. Dokl.13,6 (1972),1633-1638.
[74]B.E. Shapirovskii, Canonical sets and character. Density and weight in compact spaces, Soviet Math. Dokl.15 (1974),1282-1287.
[75]B.E. Shapirovskii, On π-character and 7r-weight of compact Hausdorff spaces, Soviet Math.Dokl.,16 (1975),999-1003.
[76]V.V. Uspenskii, The Mal'tsev operation on countably compact spaces, Com-ment. Math. Univ. Carolin.30 (1989),395-402.
[77]V.V. Uspenskii, Topological groups and Dugundji spaces, Mat. Sb.180 (1989), 1092-1118.
[78]N.Y. Vilenkin, Theory of weakly separable topological groups, Mat. Sb.22 (1948),135-177.
[79]H.F. Wang, W. He, Notes on K-topological groups and homeomorphisms of topological groups, Bull. Aust. Math. Soc.87 (2013),493-502.
[80]H.F. Wang, W. He, Remainders and cardinal invariants, Topology Appl.164 (2014),14-23.
[81]H.F. Wang, W. He, Notes on Remainders of paratopological groups, Bull. Bel-gian Math. Soc., to appear.
[82]H.F. Wang, W. He, A note on K-spaces, Bull. Aust. Math. Soc, to appear.
[83]L.H. Xie, S. Lin, The Baire property in the remainders of semitopological groups, DOI:10.1017/S0004972712000974, Published online.
[2]A.V. Arhangel'skii, On a class of spaces containing all metric and all locally compact spaces, Mat. Sb.67 (1965),55-58 (In Russian); English transl. in: Amer. Math. Soc. Transl.92 (1970),1-39.
[3]A.V. Arhangel'skii, Relations among the invariants of topological groups and their subspaces, Russian Math. Surveys,35 (1980),1-23.
[4]A.V. Arhangel'skii, Classes of topological groups, Russian Math. Surveys 36, 3(1981),151-174.
[5]A.V. Arhangel'skii, Topological Function Spaces, Kluwer,1992.
[6]A.V. Arhangel'skii, Some connections between properties of topological groups and of their remainders. Moscow Univ. Math. Bull.54,3 (1999),1-6.
[7]A.V. Arhangel'skii, Strong disconnectedness properties and remainders in com-pactifications, Topology Appl.107 (2000),3-12.
[8]A.V. Arhangel'skii, Topological invariants in algebraic environment, Recent progress in general topology Ⅲ, by Husek and van Mill,2002,1-58.
[9]A.V. Arhangel'skii, Remainders in compactifications and generalized metriz-ability properties, Topology Appl.150 (2005),79-90.
[10]A.V. Arhangel'skii, Quotients with respect to locally compact subgroups, Hous-ton J. Math.31,1 (2005),215-226.
[11]A.V. Arhangel'skii, More on remainders close to metrizable spaces, Topology Appl.154 (2007),1084-1088.
[12]A.V. Arhangel'skii, First countability, tightness, and other cardinal invariants in remainders of topological groups, Topology Appl.154 (2007),2950-2961.
[13]A.V. Arhangel'skii, Two types of remainders of topological groups, Comment. Math. Univ. Carolin.49,1 (2008),119-126.
[14]A.V. Arhangel'skii, A study of remainders of topological groups, Fund. Math. 105 (2009),1-14.
[15]A.V. Arhangel'skii, Gδ-points in remainders of topological groups and some addition theorems in compacta, Topology Appl.156 (2009),2013-2018.
[16]A.V. Arhangel'skii, Some aspects of topological algebra and remainders of topological groups, Topology Proc.33 (2009),13-28.
[17]A.V. Arhangel'skii, The Baire properties in remainders of topological groups and other results, Comment. Math. Univ. Carolin.50,2 (2009),273-279.
[18]A.V. Arhangel'skii, A construction of a Frechet-Urysohn space, and some con-vergence concepts, Comment. Math. Univ. Carolin.51,1 (2010),99-112.
[19]A.V. Arhangel'skii, Remainders of metrizable spaces and a generalization of Lindelof E-spaces, Fund. Math.215 (2011),87-100.
[20]A.V. Arhangel'skii, Components of first-countability and various kinds of pseu-doopen mappings, Topology Appl.158 (2011),215-222.
[21]A.V. Arhangel'skii, A. Bella, Cardinal invariants in remainders and variations of tightness, Proc. Amer. Math. Soc.119,3 (1993),947-954.
[22]A.V. Arhangel'skii, A. Bella, Countable fan-tightness versus countable tight-ness, Comment. Math. Univ. Carolin.37,3 (1996),565-576.
[23]A.V. Arhangel'skii, A. Bella, On pseudocompactness of remainders of topolog-ical groups and some classes of mappings, Topology Appl. 111 (2001),21-33.
[24]A.V. Arhangel'skii, M.M. Choban, On remainders of rectifiable spaces, Topol-ogy Appl.157 (2010),789-799.
[25]A.V. Arhangel'skii, M.M. Choban, Completeness type properties of semitopo-logical groups, and the theorems of Montgomery and Ellis, Topology Proc.37 (2011),1-28.
[26]A.V. Arhangel'skii, M.M. Choban, E.P. Mihaylova, About homogeneous spaces and conditions of completeness of spaces. Math. and Education in Math. 41 (2012),129-133.
[27]A.V. Arhangel'skii, M.M. Choban, E.P. Mihaylova, About homogeneous spaces and Baire property in remainders, Math. and Education in Math.41 (2012),134-138.
[28]A.V. Arhangel'skii, M.M. Choban, E.P. Mihaylova, About topological groups and the Baire properties in remainders, Proceedings of the Forty First Spring Conference of the Union of Bulgarian Mathematicians Borovetz, April 9-12, 2012.
[29]A.V. Arhangel'skii, M.M. Choban, E.P. Mihaylova, About homogeneous spaces and the Baire properties in remainders, Proceedings of the Forty First Spring Conference of the Union of Bulgarian Mathematicians Borovetz, April 9-12,2012.
[30]A.V. Arhangel'skii, W. Just, E. Rezniczenko, P. Szeptycki, Sharp bases and weakly uniform bases versus point-countable bases, Topology Appl.100 (2000),39-46.
[31]A.V. Arhangel'skii, J.Van Mill, On topological groups with a first-countable remainder, Topology Proc.42 (2013),157-163.
[32]A.V. Arhangel'skii, E.V. Reznichenko, Paratopological and semitopological groups versus topological groups, Topology. Appl.151 (2005),107-119.
[33]A.V. Arhangel'skii, M.Tkachenko, Topological Groups and Related Structures, Atlantis Press, World Sci.,2008.
[34]D. Basile, A. Bella, About remainders in compactifications of homogeneous spaces, Comment. Math. Univ. Carolin.50,4 (2009),607-613.
[35]D. Basile, J.Van Mill, A homogeneous space of point-countable but not of countable type, Comment. Math. Univ. Carolin.48,3 (2007),459-463.
[36]D. Basile, J.Van Mill, Ohio completeness and products, Topology Appl.155 (2008),180-189.
[37]A. Bouziad, Every Cech-analytic Baire semitopological group is a topological group, Proc. Amer. Math. Soc.24,3 (1998),953-959.
[38]J. Calbrix, B. Alleche, Multifunctions and Cech-complete spaces, Topology Atlas (Proceeding of the 8th Prague Topological Symposium),1997,30-36.
[39]J. Chaber, Conditions which imply compactness in countably compact spaces, Bull. Pol. Acad. Math.,24 (1976),993-998.
[40]M.M. Coban, The structure of locally compact algebras, Serdica 18 (1992), 129-137.
[41]R. Ellis, A note on the continuity of the inverse, Proc. Amer. Math. Soc.8 (1957),372-373.
[42]R. Engelking, General Topology, Revised and completed edition, Heldermann Verlag Berlin,1989.
[43]V.V. Filippov, On perfect images of paracompact p-spaces, Soviet Math. Dok1. 176 (1967),533-536.
[44]G. Gruenhage, Generalized metric spaces, in:K. Kunen, J.E. Vaughan (Eds.), Handbook of Set-Theoretic Topology, North-Holland,1984,423-501.
[45]A.S. Gul'ko, Rectifiable spaces, Topology Appl.68 (1996),107-112.
[46]I.I. Guran, On topological groups close to being Lindelof, Soviet Math. Dokl. 23 (1981),173-175.
[47]S. Hartman, J. Mycielski, On embeddings of topological groups into connected topological groups, Colloq. Math.5 (1958),167-169.
[48]R.W. Heath and W.F. Lindgren, Weakly uniform bases, Houston J. Math.2 (1976),85-90.
[49]M. Henriksen, J.R. Isbell, Some properties of compactifications, Duke Math. J. 25 (1958),83-106.
[50]R. Hodel, Cardinal functions I, in:K. Kunen, J.E. Vaughan (Eds.), Handbook of Set-Theoretic Topology, North-Holland,1984,3-61.
[51]I. Juhasz, Z. Szentmiklossy, Convergent free sequences in compact spaces, Proc. Amer. Math. Soc.116,4 (1992),1153-1160.
[52]Y. Kodama, K. Nagami, General Topology, Iwanami Shoten, Tokyo,1974.
[53]N. Lasnev, Closed images of metric spaces, Dokl. Akad. Nauk. SSSR,170 (1966),505-507.
[54]F.C. Lin, Local properties on the remainders of the topological groups, Kodai Math. J.34,3 (2011),505-518.
[55]F.C. Lin, S.X. Lin, About remainders in compactifications of paratopological groups, ArXiv:1106.3836v1 [Math. GN] 20 June 2011.
[56]F.C. Lin, R.X. Shen, On rectifiable spaces and paratopological groups, Topol-ogy Appl.158 (2011),597-610.
[57]P. Li, L. Mou and S. Wang, Notes on questions about spaces with algebraic structures, Topology Appl.159 (2012),3619-3623.
[58]S. Lin, Generalized metric spaces and mappings (lnd edition), China Science Press, Beijing,1995 (in Chinese).
[59]C. Liu, Remainders in compactifications of topological groups, Topology Appl. 156 (2009),849-854.
[60]C. Liu, Metrizablity of paratopological (semitopological) groups, Topology Appl.159 (2012),1415-1420.
[61]C. Liu, S. Lin, Generalized metric spaces with algebraic structures, Topology Appl.157 (2010),1966-1974.
[62]V.I. Malykhin, G. Tironi, Weakly Frechet-Urysohn and Pytkeev spaces, Topol-ogy Appl.104 (2000),181-190.
[63]K. Nagami, Σ-spaces, Fund. Math.65 (1969),169-192.
[64]P.J. Nyikos, Metrizability and the Frechet-Urysohn property in topological groups, Proc. Amer. Math. Soc.83 (1981),793-801.
[65]B.A. Pasynkov, Almost-metrizable topological groups, Dokl. Akad. Nauk SSSR 161 (1965),281-284.
[66]L.X. Peng and Y.F. He, A note on topological groups and their remainders, Czechoslovak Math. J.137,62 (2012),197-214.
[67]V.I. Ponomarev, Axioms of countably and continous mappings, Bull. Polon. Sci. Ser. Math. Astr. Phys.8 (1960),127-133.
[68]V.I. Ponomarev, On metrizability of Lindelof p-spaces with a point-countable base, Dokl. Akad. Nauk SSSR 174 (1967),1274-1277.
[69]L.S. Pontryagin, Continuous Groups (Third edition), Nauka, Moscow,1973.
[70]E.G. Pytkeev, On maximally resolvable spaces, Proc. Steklov Inst. Math.4 (1984),225-230.
[71]E.A. Reznichenko, Extensions of functions defined products of pseudocom-pact spaces and continuity of the converse in pseudocompact groups, Topology Appl.59 (1994),233-244.
[72]W. Roelke, S. Dierolf, Uniform Structures on Topological Groups and Their Quotients, McGraw-Hill, New York,1981.
[73]B.E. Shapirovskij, On separability and metrizability of spaces with Suslin's condition, Soviet Math. Dokl.13,6 (1972),1633-1638.
[74]B.E. Shapirovskii, Canonical sets and character. Density and weight in compact spaces, Soviet Math. Dokl.15 (1974),1282-1287.
[75]B.E. Shapirovskii, On π-character and 7r-weight of compact Hausdorff spaces, Soviet Math.Dokl.,16 (1975),999-1003.
[76]V.V. Uspenskii, The Mal'tsev operation on countably compact spaces, Com-ment. Math. Univ. Carolin.30 (1989),395-402.
[77]V.V. Uspenskii, Topological groups and Dugundji spaces, Mat. Sb.180 (1989), 1092-1118.
[78]N.Y. Vilenkin, Theory of weakly separable topological groups, Mat. Sb.22 (1948),135-177.
[79]H.F. Wang, W. He, Notes on K-topological groups and homeomorphisms of topological groups, Bull. Aust. Math. Soc.87 (2013),493-502.
[80]H.F. Wang, W. He, Remainders and cardinal invariants, Topology Appl.164 (2014),14-23.
[81]H.F. Wang, W. He, Notes on Remainders of paratopological groups, Bull. Bel-gian Math. Soc., to appear.
[82]H.F. Wang, W. He, A note on K-spaces, Bull. Aust. Math. Soc, to appear.
[83]L.H. Xie, S. Lin, The Baire property in the remainders of semitopological groups, DOI:10.1017/S0004972712000974, Published online.