Definition of an open set confusion How is the definition of an open set related to the definition of a set being open in another set?
Saying $S \subseteq D$ is open in $D$ means that $S$ open in the subspace topology on $D$. That is, $S = U \cap D$ for some open set $U \subseteq \mathbb{R}^n$. But how is this related in any way to S being an open set definition?
I cannot find any definition else online for being open in another set which is weird so any explanation or links would be appreciated.

If you know that functions induce topologies, there is a short cut we can take. We can characterize the subspace topology as follows: let $S\subseteq D$. Then, there is a $\textit{set}$ map $i:S\to D$ which is just the inclusion; that is, it sends $s\in S$ to $s$ in $D$. We can check that your definition of an open set in the subspace topology coincides exactly with the claim that $i$ is continuous. More precisely, the open sets in $S$ are $\textit{defined}$ to be those that are inverse images of open sets in $D:\ A$ is open in $S$ if and only if there exists a open set $B$ in $D$ such that $i^{-1}(B)=A$. And since $i^{-1}(B)=B\cap S$, the two definitions coincide. 
