Abstract: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.
Keywords: remainder; compactification; topological group; normal space
DOI: DOI 10.14712/1213-7243.2015.166
AMS Subject Classification: 54D35 54D40 54A25