Having trouble understanding Open Covers

[jimmypoopins]

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.

Thanks for any help you guys can provide.

 

[matticus]

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)

 

[jimmypoopins]

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

 

[matticus]

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.

 

[jimmypoopins]

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]

If your cover is just
{ (-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.

 

[jimmypoopins]

If your cover is just
{ (-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]

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 {X_i}, where {X_i} is a cover of (a,b) is a SUBset of {X_i} which also covers (a,b).

 

[jimmypoopins]

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