Notation: $\mathbb R-0$ is the same as $\mathbb R-\{0\}$.
For example consider the dense set $\mathbb Q$ as a subspace of the topological group $(\mathbb{R}-0,\cdot)$. The cosets $r\mathbb Q$ are all dense in $\mathbb R-0$ for $r\in\mathbb R-0$. The purpose why I consider a generalization of this elementary statement is to see if it really relies on the order structure of the real line. Thus consider a topological group $(G,\cdot)$ and a dense subspace $H$ of $G$. Are all the cosets of $H$ in $G$ dense?(I think there should be many counterexamples, so what topological or algebraic property for $G$ similar to $\mathbb R$ can ensure the cosets are dense?)
This is always true. For any $g\in G$, left multiplication by $g$ is a homeomorphism $L_g:G\to G$ (since its inverse is left multiplication by $g^{-1}$). Thus if $H$ is dense in $G$, so is its image $L_g(H)=gH$.