Example

If $X=\overline{\mathbb{R}}$ is equipped with the Standard topology and $Y=[0,1)$, then $A$ is relatively Open in $Y$ if and only if $$\forall a \in A,\exists\epsilon>0\ s.t. \ \begin{cases} (a-\epsilon,a+\epsilon)\subseteq A&\text{if }a>0 \\ [0,\epsilon)\subseteq A&\text{if }a=0 \end{cases}$$