Q1: An open interval in $ \mathbb R^1 $ is not open in $ \mathbb R^2 $, then what is it?
Q2: In Cantor intersection theorem, if each Q is were non empty open set then would the intersection be empty or non-empty?

Question

anonymous · Answer

You're asking how to classify a line segment that doesn't contain its endpoints if it's embedded in R2?

experimentx · Answer

kinda yes!! http://en.wikipedia.org/wiki/Open_set#Open_and_closed_are_not_mutually_exclusive

anonymous · Answer

Are you okay with why it's definitely not open?

experimentx · Answer

it does not contain it's end point in R^1, but it's an open set in R^1, my book says the the interval is no longer open set in R^2 because it cannot contain open two ball in R^2 space inside it.

experimentx · Answer

i am guessing it to be neither open nor closed set.

anonymous · Answer

I would agree.  For a set to be open, there needs to exist a tiny little interval around every point in the set such that all points in the interval also belong to the set.  in R1, that looks like this:
|dw:1356810565391:dw|

anonymous · Answer

But in R2, you need a circle, not just an interval:

|dw:1356810632181:dw|