mathematics - Why is physical space equivalent to $mathbb{R}^3$?

Why is physical space equivalent to $\mathbb{R}^3$, as opposed to e.g. $\mathbb{Q}^3$?

I am trying to understand what would be the logical reasons behind our assumption that our physical space is equivalent to $\mathbb{R}^3$ or 'physical straight line' is equivalent to $\mathbb{R}$ .

The set of reals $\mathbb{R}$ is a basically an algebraically constructed set, which is nothing but the completion of $\mathbb{Q}$, the set of rationals. For reference see here http://en.wikipedia.org/wiki/Construction_of_the_real_numbers. Now my question is what is the reason behind our approximation of the physical space by this abstract set. Why is this approximation assumed to be most suitable or good approximation?