If $A$ and $B$ are connected subsets of $\mathbb{R}^p$, give examples
to show that $A\cup B$, $A\cap B$, $A\setminus B$ can be either
connected or disconnected.
The solution for the problem is:
Let $A=[0,2]$ and $B=[1,3]$. Then $A\cup B = [0,3]$, $A\cap B=[2,3]$, and $A\setminus B=[0,1)$ are all connected. On the other hand, let $A=[0,1]$ and $B=[2,3]$. Then $A\cup B = [0,1]\cup [2,3]$ is disconnected. If $A=[0,3]$ and $B=[1,2]$, then $A\setminus B=[0,1)\cup (2,3]$ is disconnected.
I do not understand why this is the case?
Best Answer
Except for fixing $A\cap B=[1,2]$ as MITjanitor points out in the comments, the disconnected examples are correct, except I've noticed you have not given an example for $A\cap B$.
First, lets answer the question of why the sets you gave are disconnected. By definition, all we need to do is find two disjoint open sets $U$ and $V$ where $A\subseteq U$ and $B\subseteq V$. For this, take $U=(-1/2,3/2)$ and $V=(3/2,7/2)$. Can you check these contain $A$ and $B$, respectively?
For the case $A\cap B$, let $$A=\left\{(x,y)\in\Bbb R^2:\, x^2+y^2=1,\ y\geq0\right\}$$ (the top half of the unit circle), and let $$B=\left\{(x,y)\in\Bbb R^2:\,x^2+y^2=1,\ y\leq0\right\}.$$Their intersection gives $$A\cap B=\{(-1,0)\}\cup\{(1,0)\},$$and these points can be separated by open balls centered at each point with radius $1/2$.