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## 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.

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.