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Thread: proof for open set

  1. #1
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    proof for open set

    Given the set S={(x,y): |x|<1 , |y|<1} prove that is open in $\displaystyle R^2$.

    Using the definition of the open set we have:

    S is open in $\displaystyle R^2$ iff for all $\displaystyle (x_{1},y_{1})\in S$ ,there exists an ε>0 such that ,for all $\displaystyle (x_{2},y_{2})$:


    $\displaystyle (x_{2},y_{2})\in B((x_{1},y_{1})\epsilon)\Longrightarrow(x_{2},y_{2 })\in S $.


    .......................................OR......... ..................................................


    if $\displaystyle |x_{1}|<1,|y_{1}|<1$ ,there exists an ε>0 such that:


    $\displaystyle \sqrt{(x_{1}-x_{2})^2 + (y_{1}-y_{2})^2}<\epsilon\Longrightarrow |x_{2}|<1,|y_{2}|<1$

    What is the epsilon ,so that the implication can be satisfied ??

    That is as far i can go.

    Any suggestions??
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  2. #2
    Senior Member Dinkydoe's Avatar
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    Choose a point $\displaystyle x:=(x_0,y_0)\in S$

    Then the maximum $\displaystyle \epsilon > 0$ such that $\displaystyle B^o(x, \epsilon)\subset S$ is:

    $\displaystyle \epsilon_{\max} = \min\left\{|1-x_0|, |1-y_0|\right\}$.

    But, if you want to show the given square $\displaystyle S$ is open, you need to show that $\displaystyle \epsilon_{\max}> 0$ for all $\displaystyle (x_0,y_0)\in S$
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  3. #3
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    Quote Originally Posted by Dinkydoe View Post
    Choose a point $\displaystyle x:=(x_0,y_0)\in S$

    Then the maximum $\displaystyle \epsilon > 0$ such that $\displaystyle B^o(x, \epsilon)\subset S$ is:

    $\displaystyle \epsilon_{\max} = \min\left\{|1-x_0|, |1-y_0|\right\}$.

    But, if you want to show the given square $\displaystyle S$ is open, you need to show that $\displaystyle \epsilon_{\max}> 0$ for all $\displaystyle (x_0,y_0)\in S$

    So we have:


    $\displaystyle \sqrt{(x_{1}-x_{2})^2 + (y_{1}-y_{2})^2}<\epsilon=$min{$\displaystyle (1-|x_{1}|),(1-|y_{1}|)$}$\displaystyle \Longrightarrow |x_{2}|<1,|y_{2}|<1$.

    But how do we get that : $\displaystyle |x_{2}|<1,|y_{2}|<1$ now??
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  4. #4
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    Quote Originally Posted by alexandros View Post
    So we have:


    $\displaystyle \sqrt{(x_{1}-x_{2})^2 + (y_{1}-y_{2})^2}<\epsilon=$min{$\displaystyle (1-|x_{1}|),(1-|y_{1}|)$}$\displaystyle \Longrightarrow |x_{2}|<1,|y_{2}|<1$.

    But how do we get that : $\displaystyle |x_{2}|<1,|y_{2}|<1$ now??
    If $\displaystyle \left(x_1,y_1\right)\in \mathcal{B}\left(x_0,y_0\right); \varepsilon) $ then $\displaystyle \left| {x_1 } \right| \leqslant \left| {x_1 - x_0 } \right| + \left| {x_0 } \right| < \left( {1 - \left| {x_0 } \right|} \right) + \left| {x_0 } \right| = 1$.
    Likewise $\displaystyle \left| {y_1 } \right| < 1$

    That proves that $\displaystyle \mathcal{B}\left((x_0,y_0); \varepsilon\right) \subseteq \mathcal{S}$.
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