A study of remainders of topological groups
Some duality theorems relating properties of topological groups to properties of their remainders are established. It is shown that no Dowker space can be a remainder of a topological group. Perfect normality of a remainder of a topological group is consistently equivalent to hereditary Lindelöfness of this remainder. No $L$-space can be a remainder of a non-locally compact topological group. Normality is equivalent to collectionwise normality for remainders of topological groups. If a non-locally compact topological group $G$ has a hereditarily Lindelöf remainder, then $G$ is separable and metrizable. We also present several other criteria for a topological group $G$ to be separable and metrizable. Two of them are of general nature and depend heavily on a new criterion for Lindelöfness of a topological group in terms of remainders. One of them generalizes a theorem of the author [Topology Appl. 150 (2005)] as follows: a topological group $G$ is separable and metrizable if and only if some remainder of $G$ has locally a $G_\delta $-diagonal. We also study how close are the topological properties of topological groups that have homeomorphic remainders.