# open covers

• February 23rd 2011, 01:03 PM
mremwo
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!
• February 23rd 2011, 01:07 PM
Plato
Quote:

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

If $\displaystyle n\in\mathbb{N}$ let $\mathscr{O}_n=(n-0.1,n+0.1).$
• February 23rd 2011, 01:27 PM
mremwo
Why does that work/ how did you come up with this?
• February 23rd 2011, 01:44 PM
Plato
Quote:

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?
• February 23rd 2011, 02:01 PM
seamstress
I was looking at this as well. Thank you!
• March 3rd 2011, 12:02 PM
Hartlw
Quote:

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

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