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Math Help - question in real analysis

  1. #1
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    question in real analysis

    is an irrational numbers a connected set ? if true proof it .
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  2. #2
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    Consider this: between any two irrational numbers there exist a rational number.
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  3. #3
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    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 \{(\mathbb{R} - \mathbb{Q})\cap (- \infty, 2)\}\cup \{(\mathbb{R} - \mathbb{Q})\cap (2, \infty)\} . Now recall that a set A is connected iff the only sets that are both open and closed (in A) are the empty set and A itself.

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