I don't know where to begin on this problem, because I don't understand open covers that great. I feel like if someone were to help get me started and guide me through it, it would be good practice for me.
I am being asked to exhibit an open cover of N (natural numbers) that has no finite subcover.
Any help and hints would be greatly appreciated. Thanks!
Another interesting example of an open cover for N={1,2,3...} is the collection of open intervals (sets) C = (0,n), n N.
Proof: Given n N, n (0,n+1).
If N includes 0, C = (-.5,N), or C = (-a,n+b), n N, any positive a and b, cover N.
A finite cover exists only for a closed and bounded set, and N is not bounded (basic theorem).
ref: Taylor, Sec 2-5, "Covering Theorems"
note: (0,n) = {x:0<x<n}