# Having trouble understanding Open Covers

## Main Question or Discussion Point

Hey all, i'm hoping someone can help me understand this example in my book. I'm pretty bad at analysis, so explaining things as elementary as possible would be nice.

Example: Consider the open interval (0,1). For each point x in (0,1) let O_x be the open interval (x/2, 1). Taken together, the infinite collection {O_x : x in (0,1)} forms an open cover for the open interval (0,1). Notice, however, that it is possible to find a finite subcover. Given any proposed finite subcollection
{O_x_1, O_x_2, ..., O_x_n)
set x' = min{x_1, x_2, ..., x_n) and observe that any real number y satisfying 0 < y <= x'/2 (that symbol is less than or equal to) is not contained in the union from i=1 to n O_x_i.

I understand that the infinite collection of open intervals forms an open cover for the open interval (0,1), but i don't understand why you can't come up with a finite amount of open intervals, like for example, (-1,2/3) and (1/3,2) to cover the open interval (0,1). Is it because -1 and 2 aren't contained in (0,1)? The definition of open cover is a little vague in my book.

it says it's impossible to find a finite SUBcover. they're saying there is no finite subset of their proposed cover that covers (0,1)

ohhh, okay, i get it. so a SUBcover of (a,b) implies it's a collection of SUBsets of (a,b)?

no, not quite. given a cover C, a subcover is a subset of that cover that still covers the set. they give an example of a cover ({O_x : x in (0,1)}). they're saying there is no FINITE subset of this cover that still covers (0,1). you can't have a subcover without a cover.

does my example of (-1, 2/3) and (1/3, 2) count as an open subcover of the cover (-1, 2) to cover the interval (0,1), then?

CompuChip
Homework Helper
{ (-1, 2) }
(is this even a cover, wouldn't it have to be { (-1, 2) intersected with (0, 1) } ?
then no, because (-1, 2/3) and (1/3, 2) are both not in the cover.
If you'd consider { (-1, 2), (-1, 2/3), (1/3, 2) } a valid cover (despite the fact that the intervals lie outside (0, 1)) then the answer would be "yes", though.

{ (-1, 2) }
(is this even a cover, wouldn't it have to be { (-1, 2) intersected with (0, 1) } ?
then no, because (-1, 2/3) and (1/3, 2) are both not in the cover.
If you'd consider { (-1, 2), (-1, 2/3), (1/3, 2) } a valid cover (despite the fact that the intervals lie outside (0, 1)) then the answer would be "yes", though.
i don't know whether this is a cover or not, and that's one of the reasons i posted this thread, because i'm having difficulty understanding what a cover and subcover actually are.

HallsofIvy
Homework Helper
i don't know whether this is a cover or not, and that's one of the reasons i posted this thread, because i'm having difficulty understanding what a cover and subcover actually are.
Yes, {(1,2)} is a "cover" of (0, 1) because every point of (0, 1) is in some set in {(-1,2)}. In fact, since there is only one set in that cover, it is true because (0, 1) is a subset of (-1, 2).
{(1/n, 1)} is a also a cover of (0,1) because, for any x in (0,1), that is, any x such that 0< x< 1, we can find n such that 1/n< x. (Choose integer n> x. Then x< 1/n.) There is no finite subcover because then there would have to be a largest N. choose x< 1/N for that largest N. It is not in any of the intervals.

ohhh, okay, i get it. so a SUBcover of (a,b) implies it's a collection of SUBsets of (a,b)?
No, a SUBcover of {Xi}, where {Xi} is a cover of (a,b) is a SUBset of {Xi} which also covers (a,b).

thanks for all the replies, guys. i think i got it now :)