Suppose we are asked to prove that the quotient space $\mathbb R/\mathbb Z$ of $\mathbb R$ equipped with the quotient topology is compact. Has this question provided enough information for us to answer it? Do we not need to know the topology given to $\mathbb R$? I ask this because we can attach a wide variety of topologies to $\mathbb R$.
You may generally assume that any $\Bbb R^n$ has the usual (Euclidean) topology unless some other topology is explicitly specified or made very clear by the context.