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Give an example to show that Cantor's Intersection Theorem would not be true if compact sets were replaced by closed sets.


Compact set is closed and bounded, so what Im going to find is something that is closed but not bounded.

By the Cantor theorem, which said the a decreasing sequence of non-empty compact subset will have a non-empty intersection.

What I was trying to approach is how two sets just "touched" and as if they have not bounded, two sets "touched" nothing. But I cant give an example for that

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up vote 4 down vote accepted

Consider the intersection of all sets of the form $[n,\infty)$, where $n$ ranges over the positive integers.

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I had intended $n$ to range over the positive integers, and have modified the post to make that clear. Over all reals is fine too, but I wanted the intersection to be a countable intersection, since that gives a stronger result. –  André Nicolas Mar 10 '13 at 22:56
    
but (my concept isn't very good), isnt the set $[0,\infty)$ will cover all the other sets? or say intersect with every $n>0$? –  Paul Mar 10 '13 at 22:56
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It is an intersection, not a union. Let $U_n=[n,\infty)$. Then we are looking at the nested sequence $U_1,U_2,U_3,\dots$. –  André Nicolas Mar 10 '13 at 22:58
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