3 problems

Let $X$ be a metrizable space, and let $F_4(X)$ denote the subspace of the free topological group $F(X)$ consisting of words of reduced length at most $4$ with respect to the free…

Let $X$ be a Tychonoff topological space, let $F_{\mathrm{SIN}}(X)$ be the free balanced topological group on $X$, and, for a Hilbert space ${\mathcal H}$, let $U({\mathcal H})_u$…

Countability conjecture. If $X$ contains no non-trivial convergent sequences, then $G(X)$ is $snf$-countable.