From Complex Analysis by Ahlfors, pg 54,
... If we regard the closed interval $0\leqslant x\leqslant1$ as a subspace of $\mathbb R$ then the semi closed interval $0\leqslant x<1$ is relatively open but not open in $\mathbb R$.
How is it not open in $\mathbb R$?
I am finding it difficult to understand the concept of relatively open sets a simple example would help.
The set $[0,1)$ is not open in $\mathbb R$ because $0\in[0,1)$ but there is no $\varepsilon>0$ such that $(-\varepsilon,\varepsilon)\subset[0,1)$.
However, it is relatively open in $[0,1]$, since $[0,1)=(-1,1)\cap[0,1]$ and $(-1,1)$ is an open subset of $\mathbb R$.