Cantor's completeness principle 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? 

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.
