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Thread: connected sets

  1. #1
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    Post connected sets

    Hi, I'm having trouble with this:
    Prove that the set R2\ Q2 is connected
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  2. #2
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    Recall that $\displaystyle \left( {a,b} \right) \notin \mathbb{R}^2 \backslash \mathbb{Q}^2 \Rightarrow \quad a \notin \mathbb{Q}^2 \vee b \notin \mathbb{Q}^2 $.
    So suppose that $\displaystyle \left\{ {\left( {a,b} \right),\left( {c,d} \right)} \right\} \notin \mathbb{R}^2 \backslash \mathbb{Q}^2 $.
    Construct a polygonal path from $\displaystyle (a,b) \mbox{ to } (c,d)$ which is a subset of $\displaystyle
    \mathbb{R}^2 \backslash \mathbb{Q}^2 $.
    There are several cases. I will help with one.
    If $\displaystyle a \notin \mathbb{Q} \wedge d \notin \mathbb{Q}$ there is a linear path from $\displaystyle (a,b) \mbox{ to } (a,d)$: $\displaystyle \alpha _1 = \left\{ {\left( {a,t(d - b) + b} \right)} \right\};\;0 \leqslant t \leqslant 1$.
    And there is a linear path from $\displaystyle (a,d) \mbox{ to } (c,d)$: $\displaystyle \alpha _2 = \left\{ {\left( {t(c-a)+a,d} \right)} \right\};\;0 \leqslant t \leqslant 1$.
    Putting the two together we have a polygonal path from $\displaystyle (a,b) \mbox{ to } (c,d)$ which is a subset of $\displaystyle \mathbb{R}^2 \backslash \mathbb{Q}^2 $.
    There are other cases. But in each you can show pathwise connectivity.
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