Consider $C$ is the open cover { $(-n,n): n$$\in$ lN}. I would like to cconstruct a locally refinement of $C$. What I considered was $C'$={ $(n,n+2): n$$\in$ lN}. This covers the real line, is locally finite and every $(n,n+2)$ is in $(-n-3,n+3)$ $\in C$, so it is a refinement.
I just wanted to know if what I did is correct. Thank you in advance!