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

[Ka Yan]

There is a theorem: If {En} is a sequence of closed, nonempty and bounded sets in a complete metric space X, if En\supsetEn+1, and if lim diam En = 0, then \capEn 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.

 

[CompuChip]

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.

 

[tiny-tim]

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!

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