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Math Help - Metric Space , Close and open

  1. #1
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    Question Metric Space , Close and open

    We have a, b ∈ R \ Q, a < b.
    A = {x ∈ Q: a < x < b}

    Show that A is clopen(open and close) in Q

    i have being trying to solve this problem in the last 2hours

    i need some hints or examples

    thank you in advance
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  2. #2
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    Quote Originally Posted by donsmith View Post
    We have a, b ∈ R \ Q, a < b.
    A = {x ∈ Q: a < x < b}

    Show that A is clopen(open and close) in Q

    i have being trying to solve this problem in the last 2hours

    i need some hints or examples

    thank you in advance
    Assuming you give \mathbb{Q} the topology inherited from \mathbb{R} then open sets in \mathbb{Q} are those of the form A \cap \mathbb{Q} where A is open in \mathbb{R}. Is A_1=\{ x \in \mathbb{R} : a<x<b \} open? A_1 fails to be closed in \mathbb{R} because it lacks a and b, but is that a problem in \mathbb{Q}?
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  3. #3
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    Quote Originally Posted by Jose27 View Post
    Assuming you give \mathbb{Q} the topology inherited from \mathbb{R} then open sets in \mathbb{Q} are those of the form A \cap \mathbb{Q} where A is open in \mathbb{R}. Is A_1=\{ x \in \mathbb{R} : a<x<b \} open? A_1 fails to be closed in \mathbb{R} because it lacks a and b, but is that a problem in \mathbb{Q}?
    is a problem in Q

    I don't know how to start my proof.

    thank you for your reply
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