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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:https://www.physicsforums.com/attachments/7774I 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:

View attachment 7775

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