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  1. MHB Master
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    Peter's Avatar
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    #1
    I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ...

    I am focused on Chapter 1: Continuity ... ...

    I need help with an aspect of D&K's definition of disconnectedness/connectedness ... ...

    Duistermaat and Kolk's definition of disconnectedness/connectedness reads as follows:











    I tried to imagine the typical case of a disconnected set $ \displaystyle A$ ... there would be two open sets $ \displaystyle U, V \text{ in } \mathbb{R}^n$ ... indeed both sets could be contained in $ \displaystyle A$ and we would require, among other things that

    $ \displaystyle ( A \cap U) \cup (A \cap V) = A$

    This, I imagine, would give a situation like the figure below:













    BUT ... since $ \displaystyle U, V$ are open they cannot contain their boundary ... but if they do not contain their boundary ... I am assuming a common boundary ... how can we ever have $ \displaystyle ( A \cap U) \cup (A \cap V) = A$ ...


    Can someone explain where my informal thinking is going wrong ... since there obviously must exist disconnected sets ...


    Help will be appreciated ...

    Peter

  2. MHB Craftsman
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    #2
    Hi Peter,

    There is no reason why the boundary would be in $A$; that is the whole point of disconnected sets.

    As an example, take $A=\mathbb{R}\setminus\{0\}$. This is the disjoint union of two open sets.

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