Rudin, theorem 2.30 intuition behind could you show 1-2 real examples that is related to this theorem? 
2.30 Theorem: Suppose ⊂. A subset  of  is open relative to  if and only if =∩ for some open subset  of .

Take $\mathbb{R}=Y\subset X=\mathbb{R}^2$. Then any open set $E$ in $\mathbb{R}$ is given by a countable union of open intervals, $\Omega=\bigcup_{j\in J}(a_j,b_j)$. 
Each $(a_j,b_j)$ can be rewritten as $B\left(\frac{a_j+b_j}{2},\frac{b_j-a_j}{2}\right)\cap \mathbb{R}$, i.e. a ball in $\mathbb{R}^2$ of radius half the length of the interval, centered at the middle of the interval?
Then,
$$
E=\bigcup_{j\in J}(a_j,b_j)=\bigcup_{j\in J}B\left(\frac{a_j+b_j}{2},\frac{b_j-a_j}{2}\right)\cap \mathbb{R}\\
=\mathbb{R}\cap\bigcup_{j\in J}B\left(\frac{a_j+b_j}{2},\frac{b_j-a_j}{2}\right)
$$
and $G=\bigcup_{j\in J}B\left(\frac{a_j+b_j}{2},\frac{b_j-a_j}{2}\right)$.
It might be fun to try the above out on $\mathbb{R}^k\subset \mathbb{R}^n$, $k<n$.
