What would be an example of an open cover on (1,2) with no finite subcover?
Let $\displaystyle \mathcal{O}_n=\left(1,\, 2-\frac{1}{n+1}\right)$. Since
$\displaystyle \bigcup_{n=1}^{\infty}\mathcal{O}_n=(1,2),$
$\displaystyle \{\mathcal{O}_n\}$ is an open cover for $\displaystyle (1,2)$.
Now let $\displaystyle A=\{(a_1,b_1),(a_2,b_2), \dots , (a_m,b_m) \}$ be a finite subclass of $\displaystyle \{\mathcal{O}_n\}$. If we let $\displaystyle b=\max\{b_1,b_2, \dots, b_m\}$, then $\displaystyle b<2$ and
$\displaystyle \bigcup_{j=1}^m(a_j,b_j) \subseteq (1,b),$
but $\displaystyle (1,b)$ and $\displaystyle [b,2)$ are disjoint. Therefore, $\displaystyle A$ is not a cover for $\displaystyle (1,2).$