Nonnormality of remainders of some topological groups

Commentationes mathematicae Universitatis Carolinae

A. V. Arhangel'skii¹ 

关键词: remainder;    compactification;    topological group;    normal space;   

DOI  :  10.14712/1213-7243.2015.166

学科分类：物理化学和理论化学

来源: Univerzita Karlova v Praze * Matematicko-Fyzikalni Fakulta / Charles University in Prague, Faculty of Mathematics and Physics

【 摘 要 】

It is known that every remainder of a topological group is Lindel\"of 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 \v Cech-Stone remainder $G^*$ of $G$ is normal, then it is Lindel\"of. 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.

【 授权许可】

CC BY