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?

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

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