How are open subsets and relatively open subsets different from each other?

Question

To my understanding, if we have a set $U$ and a $A\subset U$ then $A$ is relatively open to $U$ if $\exists$ and open set $\Omega\subset U$ s.t $A=\Omega \cap U$. But also a subset of $U$ is open $\iff$ it is relatively open so would it isn't wrong to say that $\Omega = A$ so why do we need to define the idea of openness relative to a set?

Answer 1

Take for example $\mathbb{R}$ with the standard topology. $A = [0, 1)$ is not an open, but if you take $U = [0, 2]$, you see that $A$ is relatively open with respect to $U$, because $[0, 1) = [0, 2] \cap (-1, 1)$, which is the intersection of $U$ and an open subset of $\mathbb{R}$.

Answer 2

Consider $ \mathbb R$ with usual topology and $(0,1]$ with subspace topology wrt $\mathbb R$

Note that $(1/2,1]$ is open in $(0,1]$ but not open in $\mathbb R$

i.e. $(1/2,1]$ is relatively open in $(0,1]$