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Math Help - (path) connected, open, closed

  1. #1
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    (path) connected, open, closed

    Consider the set \mathbb{R}^2\( \mathbb{R} x {0})

    Can I say that the sets

    U:= {(x,y) in \mathbb{R}^2 \( \mathbb{R} x {0}):y>0}

    V:= {(x,y) in \mathbb{R}^2 \( \mathbb{R} x {0}):y<0}

    form a separation of \mathbb{R}^2\( \mathbb{R} x {0}), meaning that it is not (path) connected?

    Also, I'm not sure whether the set is open, closed, clopen...any ideas?
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  2. #2
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    Quote Originally Posted by TXGirl View Post
    Consider the set \mathbb{R}^2\( \mathbb{R} x {0})

    Can I say that the sets

    U:= {(x,y) in \mathbb{R}^2 \( \mathbb{R} x {0}):y>0}

    V:= {(x,y) in \mathbb{R}^2 \( \mathbb{R} x {0}):y<0}

    form a separation of \mathbb{R}^2\( \mathbb{R} x {0}), meaning that it is not (path) connected?

    Also, I'm not sure whether the set is open, closed, clopen...any ideas?
    In order for U,V to be a disconnection of \mathbb{R}^2 \setminus \mathbb{R}\times \{ 0 \} we require that U\cup V = \mathbb{R}^2\setminus \mathbb{R}\times \{ 0 \} and U\cap V = \emptyset with U,V open.
    The sets are indeed disjoint and do unionize to give the whole set. It thus only remains to show that they are open.
    Can you see why they are open sets?
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    In order for U,V to be a disconnection of \mathbb{R}^2 \setminus \mathbb{R}\times \{ 0 \} we require that U\cup V = \mathbb{R}^2\setminus \mathbb{R}\times \{ 0 \} and U\cap V = \emptyset with U,V open.
    The sets are indeed disjoint and do unionize to give the whole set. It thus only remains to show that they are open.
    Can you see why they are open sets?
    Yes, because we are dealing with strict inequality, the epsilon-neighborhood of every point in say, U is also in U (analogously for V).

    Regarding open/close of \mathbb{R}^2 \setminus \mathbb({R}\times \{ 0 \}) in \mathbb{R}^2, am I correct in the following:

    \mathbb{R}^2 \setminus \mathbb({R}\times \{ 0 \}), being the union of open sets, is open in \mathbb{R}^2.

    and because \mathbb({R}\times \{ 0 \}) is in the boundary of \mathbb{R}^2 \setminus \mathbb({R}\times \{ 0 \}) but not in \mathbb{R}^2 \setminus \mathbb({R}\times \{ 0 \}), it follows that \mathbb{R}^2 \setminus \mathbb({R}\times \{ 0 \}) is not closed?

    I'm really not certain about the "NOT closed" part...
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