I had a discussion with a friend that say that
transcendental numbers can be better approximated by rationals than algebraic (irrational) numbers, so, is some sense, they are more ''near'' to rationals that the algebraic numbers.
I don't understand in what sense this statement can be true (and I suspect that it is false).
I know that the way to estimate how a irrational number is approximated by rational is the irrationality measure, and that this measure is $1$ for rational numbers, $2$ for algebraic numbers, $\ge 2$ for transcendental numbers and $\infty$ for Liouville numbers.
I interpret this as the Liouville numbers are the numbers better approximated by rationals (correct?) and we know that these numbers are transcendental. But I also know that the set of Liouville numbers has measure $0$, so almost all transcendental numbers have irrationality measure $\ge 2$ but $< \infty$. But we cannot prove that almost all have irrationality measure $>2$, so we cannot say that thay are generally better approximated by rationals. Or there is some proof?
Anyway, i think that the idea of ''near to'' cannot be applied in this situation since it is obviously false in standard topology of reals and I don't see a topology in which it can be true. Or there is someone?