My real-analysis text gave the following defintion:
Let U be a subset of E. U is open relative to E if
for $\forall t \in U$, $\exists \epsilon$ such that $N_\epsilon(t) \cap E \subset U$.
Although the idea that U is open in $\mathbb R$ follows the definition,
I normally do not think about the intersection of $N_\epsilon(t) \cap \mathbb R$.
U is open if for $\forall t \in U$, $\exists \epsilon$ such that $N_\epsilon(t) \subset U$; every t is an interior point of U.
Intuitively, each point, t, is contained in a "bubble".
So, what is the significance of specifying the intersection?
Apparently, I can't think of a relative open set in terms like interior points and "bubbles".