Taking infinite (binary) product in topological groups

Question

Normally when we work in a group $G$, the pure algebraic structure doesn't allow us to make sense of infinite the infinite product $g_1g_2g_3...$ where $g_i\in G$. However, if we give $G$ a topology, this can change. A motivating example is $(\mathbb{Q},+)$, where $$1=\sum_\infty 2^{-n}$$ where we now have convergence of the infinite sum. I'm under the impression (I could be wrong) that even when talking about topological or Lie groups, we don't usually consider these sort of convergence issues. Is there a standard reference, or even name, for what these structures are?