Quotients of Primes

Abstract

This note is motivated by our answer, appearing herein as Theorem 4, to the following question posed in a communication from our colleague William Vince Grounds: Given positive real numbers a V. When X is a topological space then the least cardinality of a dense subset of X is one among several interlocking measures of the 'poverty' of the topology on X. For instance, a topological space with a countable dense subset is called separable. These traits are 'topological invariants', and are fundamental to the classification of topological spaces. It is well-known that lR under the usual topology is separable, and indeed that the denumerable set F(N) is dense in the subspace lR'. Theorem 4 states that F(P') is dense in Rll. If S is a finite subset of RN1 then F(S) is finite, and therefore F(S) fails to be dense in X for every infinite subspace X of 1R.