Proving Non-Empty Compact Sets in n-Dimensional Space

[trap]

Any help will be very gracious.

If $$\text C_{1} , C_{2} , C_{3}$$ are all non empty compact sets in $$\text R^n$$ such that $$\text C_{k+1} \subset C_{k}$$ for all k=1,2,3,..., then the set $$\text C = I_{k=1}^{\infty}C_{k}$$ is also non-empty.

 

[CarlB]

The basic problem is that there are an infinite number of $C_n$ involved. You should already know how to deal with unions of infinite numbers of open sets. So try looking at the complements of $C_n$.

Carl

 

[HallsofIvy]

The complement of a closed set is open and of course all compact sets are closed. But it is crucial that these sets be compact and don't see how looking at their complements will include that.

If you were allowed to use the "Finite Intersection Propery"- If the intersection of any finite subset of a collection of compact sets is non-empty, then the intersection of all of them is non-empty- this would be trivial. However, I suspect that the whole point of this is to prove a relatively simple version of that.

Try this: Let {U_α} be an open cover for C₁. Then, since every C_n is a subset of C₁, it is also an open cover for C_n. Since each C_n is a compact, there exist a finite sub-cover.
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