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:

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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:

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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

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.