5.3. Connected and Disconnected Sets
Is the set of rational numbers connected or disconnected ? How about the irrationals ?Both sets are disconnected. The idea is simple: The set of rational numbers can be 'split' at any irrational number, while the set of irrationals can be 'split' at any rational number. Can you provide the details ? Note that neither set is open, so that the more complicated definition of disconnected must be used.
In fact, both of these sets are totally disconnected.