What Happens to the Intersection of Open Sets in Incomplete Spaces?

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In summary, the conversation discusses a theorem that states if a sequence of closed, nonempty and bounded sets in a complete metric space X satisfies certain conditions, then their intersection will consist of exactly one point. The person asks what would happen if the sets were not closed or if X was not a complete space, but still satisfied the other conditions. They also wonder what would happen if X was the set of rational numbers. The other person suggests finding a proof for the theorem and examining which assumptions are used. They also mention finding an example of open sets with an empty intersection and question whether the set of rationals is complete.
  • #1
Ka Yan
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There is a theorem: If {En} is a sequence of closed, nonempty and bounded sets in a complete metric space X, if En[tex]\supset[/tex]En+1, and if lim diam En = 0, then [tex]\cap[/tex]En consists exactly one point.

And what I'm asking is that, if either the sets were not closed or X was not a complete space (but not both), and all other condictions are still satisfied, then what will follow? And if I let X be the rational set, for instance, what will I get. And could you explain it?

Thks.
 
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  • #2
Do you have the proof for this theorem? Then you could just scan through it and scrutinize each step to see which assumption(s) are used.
 
  • #3
Hi, Ka Yan!

You should be able to find a simple example of open sets (on a plane, say) whose intersection is empty.

… there you go! :smile:

(and: hint: are the rationals complete? if not, why not?)
 

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