The densest subset of real numbers is the set of fractions?

Answer 1

Your question is ill-posed. I have not come across a comparison dense-denser-densest.

The term "dense" is a topological property of a set:

A set A is dense in a set B, if for all y in B, there is an open set O of B, such that O and A have nonempty intersection.

The rational numbers are indeed dense in the set of real numbers with the standard topology. An open set containing a real number contains always a rational number.

Another way of saying it is that every real number can be approximated to any precision by rational numbers.

There are denser sets, if you are willing to consider more elements.

Suppose you construct a set consisting of the rational numbers plus all algebraic numbers. The set of algebraic numbers is also countable, but adding them, makes it obviously easier to approximate real numbers.

Can you perhaps construct a set less dense than the set of rational numbers?

Suppose we take the set of rational numbers without the element 0. Is this set still dense in the real numbers? Yes, because 0 can be approximated by 1/n, n>1.

In fact, you can remove finite number of rational numbers from the set of rational numbers and the resulting set will still be dense in the set of the real numbers.