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Math Help - Open sets Proof

  1. #1
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    Open sets Proof

    A) Show that the union of two nonempty open sets is open.

    I have this theorem to use: A set D is open iff it contains no point of its boundary.

    Can someone show have to give a solid proof of A...

    Thanks in advance for any help
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  2. #2
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    Here is what I think:

    Say I have open sets X and Y, then for  \forall x \in X , \forall y \in Y , I can find an open ball centered at them such that the entire ball is in X and Y, respectively.

    Well, then, if you pick any point in  X \cup Y , that that point is either in X or Y, well, then, you can draw another open ball that is contained in either X or Y.

    So if you use radius  \epsilon , \delta for the points in X and Y, you can just use radius  min \{ \epsilon , \delta \}

    Hope this helps.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by jzellt View Post
    A) Show that the union of two nonempty open sets is open.

    I have this theorem to use: A set D is open iff it contains no point of its boundary.

    Can someone show have to give a solid proof of A...

    Thanks in advance for any help
    Is this a topological space or a metric space? If it is the former than this is the definition. Otherwise, let \left\{O_j\right\}_{j\in\mathcal{J}} be an arbitrary class of open sets in a metric space X. Let x\in\bigcup_{j\in\mathcal{J}}O_j be arbitrary. Since x\in O_k for some k and O_k is open there exists a \delta>0 such that B_{\delta}(x)\subseteq O_k\subseteq\bigcup_{j\in\mathcal{J}}O_j and we are finished.
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