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Thread: Finding Open Cover

  1. #1
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    Finding Open Cover

    Let $\displaystyle A = \mathbb{Q} \cap [0,1] \subset \mathbb{R} $. Give an example of an open cover of $\displaystyle A $ which has no finite subcover.

    I was thinking, since it specifies exactly that $\displaystyle A \subset \mathbb{R} $, then do our open sets (that are in the open covers) have to be open in $\displaystyle \mathbb{R} $, or can they be open in $\displaystyle \mathbb{Q} $?

    For example, if $\displaystyle x $ is any rational number in $\displaystyle [0,1] $ then the set $\displaystyle \{ x \} $ is open in $\displaystyle \mathbb{Q} $, and the collection of all sets {x} such that x is a rational number in [0,1] is an open cover which has no finite subcover. But is it permissible for my open sets to be open in $\displaystyle \mathbb{Q} $? Or do they have to be open in $\displaystyle \mathbb{R} $?
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  2. #2
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    No, they have to be open in R precisely because it said "$\displaystyle \subset \mathbb{R}$". A set or real numbers is compact if and only if it is both closed and bounded. A clearly is bounded so the crucial point is that it is not closed as a set of real numbers. Why, exactly,is it not closed? Use that to find your open cover.
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  3. #3
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    Yup, that's exactly what I did (before I saw your post :P ), and I found my open cover. Thanks halls.
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