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**detalosi** I am asked to argue for or against the following statement:

The interval $\displaystyle \left[0,1\right]$ is an intersection of countably many open sets of $\displaystyle \mathbb{R}$

I have the following considerations:

1) The intersection of countably many opens sets produces an open set. Since $\displaystyle \left[0,1\right]$ is closed then it can not be an intersection of countably many opens sets.

2) I consider the interval $\displaystyle \left]-\frac{1}{n},1+\frac{1}{n}\right[$. Then $\displaystyle \bigcap_{n=1}^{\infty} \left]-\frac{1}{n},1+\frac{1}{n}\right[ = \left[0,1\right] $

Based on point 1) and 2) I would conclude that $\displaystyle \left[0,1\right]$ is NOT an intersection of countably many open sets but IS an intersection of infinitely many open sets. Is the above considerations and conclusion correct? **NO ABSOLUTELY NOT!**