Interactive Real Analysis
- part of Examples 5.3.2(c):
Is the set of rational numbers connected or disconnected ? How about the
irrationals ?
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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.
BackIn fact, both of these sets are totally disconnected.