Let $G$ be a compact connected metrizable group of finite topological dimension. Is $G$ a Lie group? Or what should be equivalent in this case, does it have a faithful finite-dimensional representation?
This is something that should be well known, but it is difficult to find.
This is false. As an example, consider a solenoid $G$, which is the inverse limit of, say, $2$-fold covering maps of circles $$ ... \to S^1\to S^1\to S^1. $$ This topological group has topological dimension 1, is metrizable, compact, connected, even abelian, but is not a Lie group (it is not locally connected). The local model for this group is the product of a Cantor set with an interval.