Comparison of topologies on fundamental groups with subgroup topology viewpoint

Abstract In order to make the fundamental group (one of the most well-known invariants in algebraic topology) more useful and powerful, some researchers have introduced and studied various topologies on the fundamental group from the beginning of the 21st century onwards. In this paper, first we recall and review these topologies and then using the concept of subgroup topology, we compare these topologies in order to present some results on topologized fundamental groups.