In general, if you are given a subset of a topological space, which is itself referred to as a topological space, then it is assumed to have the subspace topology. Any subset of a topological space inherits the ambiant topology in a natural way. Let $\displaystyle \iota : S^2 \to \mathbb{R}^3$ be the inclusion map. The topology on $\displaystyle S^2$ is the topology generated by the open sets of the form $\displaystyle \iota^{-1}(U)$, where $\displaystyle U$ is open in $\displaystyle \mathbb{R}^3$.

Now it is easy to solve the problem. Show that any open cover of a set in $\displaystyle S^2$ lifts to an open cover of the same set, considered as a subset of $\displaystyle \mathbb{R}^3$. Conversely, any open cover of some subset of $\displaystyle S^2$ (considered as a subset of $\displaystyle \mathbb{R}^3$) can be considered as an open cover of the same set as a set of $\displaystyle S^2$.

By the way, the notation $\displaystyle S^n$ for the $\displaystyle n$-sphere is universal!