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intersection of the collection of open intervals

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1. Prove if F is a subset of R^n and if d(x,F)=inf(||x-z||:z in F}=0 then x belongs to F.

2.The intersection of two open sets is compact iff it is empty. Can the intersection of an infinite collection of open sets be a non-empty compact set?

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Solution Summary

This solution contextualizes metric topology. The intersection of the collection of open intervals are determined.

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1. The statement if formulated incorrectly. As stated, it is false. You have to assume that F is closed. Otherwise, consider the open interval I=(0,1) and the point x=0. The infimum of distances between x and the elements of I is zero, as can be seen by taking the elements z_n=1/n n=1, 2, 3, ... ...

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