Can anyone help me to understand why set S =(0,2) for each n in the Natural Numbers is not compact in the most basic way possible?
I know so far that S is compact iff every open cover of S contains a finite subcover.
and i also know that a Set (1/n,3) is a open cover of the set (0,2).
The collection is an open covering of .
Can you give us a finite subcover of that cover?
i don't think you can.. because you have infinite many numbers in that set (0,2-(1/n))
but a cover of that would be (0,1) ?? but this would be infinite?? if i'm not wrong..
wait i think what i said before is non sense..
couldn't the set (1/n,2), be a subcover of the set you gave??
The set is not even in the cover I gave you!
Originally Posted by pandakrap
It must be a finite collection of subsets of the given collection.
i'm sorry.. it's a bit difficult to picture this...
would (0,2-(2/n)) be a subcover of your cover (0,2-(1/n)) ??