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