# Math Help - open cover with no finite subcover

1. ## open cover with no finite subcover

What would be an example of an open cover on (1,2) with no finite subcover?

2. Let $\mathcal{O}_n=\left(1,\, 2-\frac{1}{n+1}\right)$. Since

$\bigcup_{n=1}^{\infty}\mathcal{O}_n=(1,2),$

$\{\mathcal{O}_n\}$ is an open cover for $(1,2)$.

Now let $A=\{(a_1,b_1),(a_2,b_2), \dots , (a_m,b_m) \}$ be a finite subclass of $\{\mathcal{O}_n\}$. If we let $b=\max\{b_1,b_2, \dots, b_m\}$, then $b<2$ and

$\bigcup_{j=1}^m(a_j,b_j) \subseteq (1,b),$

but $(1,b)$ and $[b,2)$ are disjoint. Therefore, $A$ is not a cover for $(1,2).$