Topology Meets Number Theory

Liouville proved the existence of a set L of transcendental real numbers now known as Liouville numbers. Erdos 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 Erd os property, that is, every real number is the sum of two numbers in L. He proved L is a dense G(delta)-subset of R and every dense G(delta)-subset of R has the Erdos property. While being a dense G(delta)-subset of R is a purely topological property, all such sets contain c Liouville numbers. Each dense G(delta)-subset of R, including L, is homeomorphic to the product N. 0 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 G(delta)-subset of R has cardinality L. Indeed, any dense G(delta)-subset of R has a chain X-m, m is an element of (0, infinity) of homeomorphic dense G(delta)-subsets such that X-m subset of X-n, for n < m, and X-n \ X-m has cardinality c. Finally, every real number r not equal 1 is equal to a(b), for some a, b is an element of L.