Open Cover of (0,1) by infinite collection of sets

Question

text given in"Calculus on manifolds" In the above text, it's mentioned that no finite collection of open cover of the form $(\frac{1}{n},1-\frac{1}{n})$, n $\in N $ can cover the interval (0,1). But if we take n = 1, it gives the inteval (1,0) which does cover the interval (0,1). So, is the statement given in the book wrong?

Answer 1

You are wrong, because for $n=1$ you get $$ \left(\frac11,1-\frac11\right)=(1,0)=\emptyset. $$ You have to consider that $$ (a,b):=\{x\in\mathbb R~:~a<x<b\}. $$ Therefore $a>b$ will give $(a,b)=\emptyset$.