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Okay my book says that a collection H of open sets is an open covering of a set S if every point in S is contained in a set F belonging to H.
Then it says that the set S = [0,1] is covered by the family of open intervals
F1 = {(x-(1/N), x +(1/N))|0< x <1}
N = positive integer
My first question is that.. Does this mean that if I pick an N and make it constant I can find an opencovering by picking arbitrary x ?
In this case how to I use the set F1 to construct the open cover of S ?
How exactly does this open covering work in this case ? According to Heine-Borel a compact set has finitely many open sets that make up it's open covering.
My second question is that if [0,1] has finitely many open sets that make up it's open covering then so should (0,1), right ? Since (0,1) is a smaller set than [0,1]. But (0,1) is not closed, however it has a supremum and infimum that are not in the open interval.Basically I want to understand how to use F1 to cover [0,1]; like what do I chose, any arbitrary N or a bunch of arbitrary x.I'm studying this stuff on my own because the Analysis course I want to take isn't going to be offered at my school this year, but I can't wait. So please have some patience .
~Regards,
Elmer
Then it says that the set S = [0,1] is covered by the family of open intervals
F1 = {(x-(1/N), x +(1/N))|0< x <1}
N = positive integer
My first question is that.. Does this mean that if I pick an N and make it constant I can find an opencovering by picking arbitrary x ?
In this case how to I use the set F1 to construct the open cover of S ?
How exactly does this open covering work in this case ? According to Heine-Borel a compact set has finitely many open sets that make up it's open covering.
My second question is that if [0,1] has finitely many open sets that make up it's open covering then so should (0,1), right ? Since (0,1) is a smaller set than [0,1]. But (0,1) is not closed, however it has a supremum and infimum that are not in the open interval.Basically I want to understand how to use F1 to cover [0,1]; like what do I chose, any arbitrary N or a bunch of arbitrary x.I'm studying this stuff on my own because the Analysis course I want to take isn't going to be offered at my school this year, but I can't wait. So please have some patience .
~Regards,
Elmer