Thread: open covers

Originally Posted by mremwo

I
I am being asked to exhibit an open cover of N (natural numbers) that has no finite subcover.

If let

Why does that work/ how did you come up with this?

Originally Posted by mremwo

Why does that work/ how did you come up with this?

What does it mean to cover a set?
What does it mean to say open cover?
What does it mean to have a sub-cover?
What all do you know about this topic?

I was looking at this as well. Thank you!

Originally Posted by mremwo

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}