Universal-homogeneous structures are generic

TL;DR

This paper demonstrates that universal-homogeneous structures, such as Hall's universal group and algebraic closures of finite fields, are generic in their respective spaces, characterized by comeager sets in a topological framework.

Contribution

It establishes that Fra"iss"e limits are comeager in a logic topology, showing their genericity among countable structures with a given age.

Findings

Hall's universal group is comeager among countable locally finite groups.

The algebraic closure of _p is comeager among countable fields of characteristic p.

Fra"iss"e limits are unique and generic in the space of structures with a fixed universe.

Abstract

We prove that the Fra\"iss\'e limit of a Fra\"iss\'e class $\mathcal C$ is the (unique) countable structure whose isomorphism type is comeager (with respect to a certain logic topology) in the Baire space of all structures whose age is contained in $\mathcal C$ and which are defined on a fixed countable universe. In particular, the set of groups isomorphic to Hall's universal group is comeager in the space of all countable locally finite groups and the set of fields isomorphic to the algebraic closure of $\mathbb F_p$ is comeager in the space of countable fields of characteristic $p$.