Constructing a closed interval from open intervals of real numbers?

Question

Is it possible to construct a closed interval, say [0,1] using only open intervals?

Ah, sorry. I should have said. Using the three set operations (union, intersection, complement) and open intervals of real numbers.

Answer 1

Tsemo Aristide's answer below shows how to answer the question assuming that you allow infinite intersections. However, even without this the answer is yes.

First, note that $[0, 1]=(-1, 2)\setminus ((-1, 0)\cup (1, 2))$ - here "$\setminus$" is set subtraction.

But $A\setminus B$ is just $A\cap B^c$ (here "$X^c$" denotes the complement of $X$).

Putting this together we can write $[0, 1]$ as the intersection of an open interval and the complement of the union of two open intervals.

Answer 2

Yes, $[0,1]=\bigcap\limits_{n\in \mathbb{N}}\mathopen]-1/n,1+1/n\mathclose[$.