- #1
jimmypoopins
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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.
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.