Topologically simple and metrizable free groups with no non-trivial NSS quotients

A topological group $G$ is said to have no small subgroup (resp. no small normal subgroup) if it admits an open neighbourhood of the identity containing no non-trivial subgroup (resp. normal subgroup) of $G$. These properties are usually denoted by NSS (and respectively NSnS). The NSS property plays an important historical role in the solution to the fifth problem of Hilbert due to Gleason, Montgomery-Zippin and Yamabe for the characterization of Lie groups. In 2019, Shakhmatov and the author proved that a free group $F$ with countably infinitely many generators admits a metric Hausdorff group topology $\mathscr{T}$ which satisfies the so-called algebraic small subgroup generating property ASSGP: for each open neighbourhood $U$ of the identity of $F$, the family of subgroups contained in $U$ algebraically generates $F$. In particular $(F,\mathscr{T})$ admits no non-trivial continuous homomorphisms to either NSS or locally compact groups, making it minimally almost periodic. In this paper, we prove that $(F, \mathscr{T})$ can be made topologically simple; namely, $(F, \mathscr{T})$ contains no closed normal subgroups other than $\{e\}$ and $F$. In particular, this implies that $F$ satisfies the no small normal subgroup (NSnS) property.