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

    I don't know how to start my proof.

    thank you for your reply
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