is an irrational numbers a connected set ? if true proof it .

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- Jan 9th 2010, 09:30 AMnice rosequestion in real analysis
is an irrational numbers a connected set ? if true proof it .

- Jan 9th 2010, 10:18 AMPlato
Consider this: between any two irrational numbers there exist a rational number.

- Jan 9th 2010, 10:50 AMh2osprey
I'm assuming you're referring to the set of irrational numbers; in that case you can write the irrationals as a union of two disjoint sets, say $\displaystyle \{(\mathbb{R} - \mathbb{Q})\cap (- \infty, 2)\}\cup \{(\mathbb{R} - \mathbb{Q})\cap (2, \infty)\} $. Now recall that a set $\displaystyle A$ is connected iff the only sets that are both open and closed (in $\displaystyle A$) are the empty set and $\displaystyle A$ itself.

From this it should clear whether the irrationals (as well as the rationals, actually) are connected or disconnected.