Definition of Connectedness .... What's wrong with my informal thinking ....?

[Math Amateur]

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: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

 

[castor28]

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.

 

[math771]

Hi Peter,

I understand your confusion. The key here is to remember that when we say "open sets U, V in R^n," we mean open sets in the Euclidean topology. In this topology, the boundary of an open set is not necessarily contained within the set itself. So, in the figure you provided, the boundary of U and V may not be contained within the sets, but they are still open sets in the Euclidean topology.

Therefore, when we say (A ∩ U) ∪ (A ∩ V) = A, we are not including the boundary points in the intersection. This is because the boundary points are not necessarily contained within the sets themselves. So, in your example, the boundary points would not be included in (A ∩ U) ∪ (A ∩ V), but this does not affect the overall result of the equation.

I hope this helps clarify the concept for you. Let me know if you have any other questions.