Question

Let X be a connected subset of real numbers. If every element of X is irrational, then the cardinality of X is

Answer 1

A subset of the real numbers is connected if and only if it is an interval. Since there exists a rational number between every two irrational numbers, a connected subset of the reals containing only irrational numbers can therefore at most have 1 element. Note that the empty set also satifies the condition