A study of remainders of topological groups

A. V. Arhangel'skii¹

¹ Department of Mathematics Ohio University Athens, OH 45701, U.S.A.

Fundamenta Mathematicae, Tome 203 (2009) no. 2, pp. 165-178

Résumé

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.

Détail

DOI : 10.4064/fm203-2-3

Keywords: duality theorems relating properties topological groups properties their remainders established shown dowker space remainder topological group perfect normality remainder topological group consistently equivalent hereditary lindel fness remainder l space remainder non locally compact topological group normality equivalent collectionwise normality remainders topological groups non locally compact topological group has hereditarily lindel remainder separable metrizable present several other criteria topological group separable metrizable general nature depend heavily criterion lindel fness topological group terms remainders generalizes theorem author topology appl follows topological group separable metrizable only remainder has locally delta diagonal study close topological properties topological groups have homeomorphic remainders