I read that a subset $S$ of a topological space $(X,\tau)$ is compact if for every open cover of $S$, there exists a finite subcover, i.e.:
$$ S \subseteq \bigcup_{i \in J}O_i $$
for a finite $J$, and where $O_i$ are open sets.. Now, are these open sets $O_i$ taken from the subspace topology of $S$? , i.e. $(S,\tau_1)$ s.t. $O_i \in \tau_1$ or are they from $\tau$?