I hope everyone who has underwent a fundamental course in Analysis must be knowing about Cantor's completeness principle. It says that in a nest of closed intervals ,the intersection of all the intervals is a single point. I hope I can get an explanation as to why in case of only closed intervals this principle holds good, why not in case of open intervals?
Cantor's completeness principle
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analysis
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0@Prime, Are you looking for examples or an explanation? – 2011-09-19
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The intersection of all the open intervals centered at $0$ is just $\{0\}$, since $0$ is the only point that is a member of all of them.
But the intersection of all the open intervals whose lower boundary is $0$ is empty. (After all, what point could be a member all of them?) And they are nested, in that for any two of them, one is a subset of the other.