On orderability of topological groups

Abstract

A necessary and sufficient condition for a topological group whose topology can be induced by a total order compatible with the group structure is given and such groups are called ordered or orderable topological groups. A separable totally disconnected ordered topological group is proved to be non-archimedean metrizable while the converse is shown to be false by means of an example. A necessary and sufficient condition for a no-totally disconnected locally compact abelian group to be orderable is also given.

Keywords

• topological groups
• orderable or ordered topological groups
• non-archimedean metrizable
• totally disconnected
• locally compact abelian groups.