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remainder; compactification; topological group; normal space
It is known that every remainder of a topological group is Lindelöf or pseudocompact. Motivated by this result, we study in this paper when a topological group $G$ has a normal remainder. In a previous paper we showed that under mild conditions on $G$, the Continuum Hypothesis implies that if the Čech-Stone remainder $G^*$ of $G$ is normal, then it is Lindelöf. Here we continue this line of investigation, mainly for the case of precompact groups. We show that no pseudocompact group, whose weight is uncountable but less than $\mathfrak c$, has a normal remainder under $\mathsf{MA}{+}\neg\mathsf{CH}$. We also show that if a precompact group with a countable network has a normal remainder, then this group is metrizable. We finally show that if $C_p(X)$ has a normal remainder, then $X$ is countable (Corollary 4.10) This result provides us with many natural examples of topological groups all remainders of which are nonnormal.
[1] Arhangel'skii A.V.: Topological Function Spaces. Math. Appl., vol. 78, Kluwer Academic Publishers, Dordrecht, 1992. MR 1144519
[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.: Two types of remainders of topological groups. Comment. Math. Univ. Carolin. 47 (2008), 119–126. MR 2433629
[4] Arhangel'skii A.V., van Mill J.: Nonnormality of Čech-Stone-remainders of topological groups. 2015, to appear in Topology Appl.
[5] Arhangel'skii A.V., Tkachenko M. G.: Topological Groups and Related Structures. Atlantis Studies in Mathematics, vol. 1, Atlantis Press, Paris, World Scientific, 2008. MR 2433295
[6] Efimov B.A.: On dyadic spaces. Soviet Math. Dokl. 4 (1963), 1131–1134. MR 0190894 | Zbl 0137.16104
[7] Engelking R.: Cartesian products and dyadic spaces. Fund. Math. 57 (1965), 287–304. DOI 10.4064/fm-57-3-287-304 | MR 0196692 | Zbl 0173.50603
[8] Fleissner W.G.: Normal Moore spaces in the constructible universe. Proc. Amer. Math. Soc. 46 (1974), 294–298. DOI 10.1090/S0002-9939-1974-0362240-4 | MR 0362240 | Zbl 0384.54016
[9] Juhász I.: Cardinal Functions in Topology – Ten Years Later. Mathematical Centre Tract, vol. 123, Mathematical Centre, Amsterdam, 1980. Zbl 0479.54001
[10] Kombarov A.P., Malyhin V. I.: $\Sigma $-products. Dokl. Akad. Nauk SSSR 213 (1973), 774–776. MR 0339073 | Zbl 0296.54008
[11] Nyikos P.J., Reichel H.-C.: Topologically orderable groups. General Topology Appl. 5 (1975), 195–204. DOI 10.1016/0016-660X(75)90020-3 | MR 0372105 | Zbl 0302.22003
[12] Raĭkov D.A.: On the completion of topological groups. Izv. Akad. Nauk SSSR 10 (1946), 513–528, (in Russian). MR 0020083 | Zbl 0061.04206
[13] Rudin M.E.: Lectures on set theoretic topology. Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, Providence, R.I., 1975, Regional Conference Series in Mathematics, No. 23. MR 0367886 | Zbl 0472.54001
[14] Tkachuk V.V.: A $C_p$-theory problem book. Springer, Cham, Berlin, 2014, xiv+583. MR 3243753 | Zbl 1325.54001
[15] Weil A.: Sur les Espaces à Structure Uniforme et sur la Topologie Générale. Hermann, Paris, 1937. Zbl 0019.18604
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