Projectively universal countable metrizable groups

TL;DR

This paper constructs a countable metrizable topological group that can generate all other countable metrizable groups as quotients, and its completion is a Polish group that can generate all Polish groups as quotients.

Contribution

It introduces a universal countable metrizable group and its Polish completion, demonstrating their quotient universality for all such groups.

Findings

Existence of a countable metrizable universal group

Its completion is a Polish universal group

Universal properties for quotients of these groups

Abstract

We prove that there exists a countable metrizable topological group $G$ such that every countable metrizable group is isomorphic to a quotient of $G$. The completion $H$ of $G$ is a Polish group such that every Polish group is isomorphic to a quotient of $H$.