Topology Meets Number Theory

Liouville proved the existence of a set L of transcendental real numbers now known as Liouville numbers. Erdős proved that while L is a small set in that its Lebesgue measure is zero, and even its s-dimensional Hausdorff measure, for each s > 0, equals zero, it has the Erdős property, that is, every real number is the sum of two numbers in L. He proved L is a dense Gδ-subset of R and every dense Gδ-subset of R has the Erdős property. While being a dense Gδ-subset of R is a purely topological property, all such sets contain c Liouville numbers. Each dense Gδ-subset of R, including L, is homeomorphic to the product Nℵ0 of copies of the discrete space N of all natural numbers. Also this product space is homeomorphic to the space P of all irrational real numbers and the space T of all transcendental real numbers. Hence every dense Gδ-subset of R has cardinality c. Indeed, any dense Gδ-subset of R has a chain Xm , m∈(0,∞) of homeomorphic dense Gδ-subsets such that Xm⊂Xn, for n < m, and Xn∖Xm has cardinality c. Finally, every real number r≠1 is equal to ab , for some a,b∈L.