Okay my book says that a collection(adsbygoogle = window.adsbygoogle || []).push({}); Hof open sets is an open covering of a setSif every point in S is contained in a setFbelonging toH.

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

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# Open coverings and Heine-Borel theorem

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