Thread: open covers

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!

Originally Posted by mremwo

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

If $\displaystyle \displaystyle n\in\mathbb{N}$ let $\displaystyle \mathscr{O}_n=(n-0.1,n+0.1).$

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 $\displaystyle \epsilon$ N.
Proof: Given n $\displaystyle \epsilon$ N, n $\displaystyle \epsilon$ (0,n+1).

If N includes 0, C = (-.5,N), or C = (-a,n+b), n $\displaystyle \epsilon$ 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}