# Existence of many finitely generated precompact subgroups of G and L_0(G)

- Speaker(s)
**Jakub Andruszkiewicz**- Affiliation
- Doctoral School of Exact and Natural Sciences UW
- Date
- April 17, 2024, 4:15 p.m.
- Room
- room 5050
- Seminar
- Topology and Set Theory Seminar

For a Polish group G one can define L_0(G) as the set of all (Borel or) Lebesgue measurable functions from [0,1] to G. This set, after identifying functions that are equal up to a set of measure zero, when given the topology of convergence in measure and the group action of pointwise multiplication, becomes a Polish group. Although in some ways G and L_0(G) may differ pretty drastically - for example, L_0(G) is always (locally) path-connected, while obviously G needs not to be - some of their properties are shared. We will show that among those is an existence of comeagerly many elements generating a precompact subgroup. In fact, we will show that if comeagerly n-tuples in G^n generate a precompact subgroup in G, then the same happens with respect to L_0(G).