3
$\begingroup$

Let $X\subset\mathbb{R}^n$ be a not connected set and $A,B\subset X$ disjoint connected components of $X$. How to prove that there are disjoint open sets $U,V\subset\mathbb{R}^n$ such that $$X\subset U\cup V$$ $$A\subset U\;,\;B\subset V$$

What happens if $X$ is compact?

Any hint would be appreciated.

  • 0
    Hint: the only connected subsets of $\Bbb{R}$ are individual points and closed, open and half-open intervals. Using this fact it is easy to prove. – 2017-01-07
  • 1
    Why is this true?. For example, If $X:=\mathbb Q_{>0}\cup\mathbb Q_{<1},$ $A:=[1/2,1]$ and $B:=(1,2),$ can you give an example of $U$ and $V$? (since $U$ is open and $1\in U$ then there is some open interval around $1$ which completely lies in $U,$ but the density of the rationals implies that such interval contains points of $B$) – 2017-01-08

1 Answers 1

2

Counterexample:

Let $n=2,\ A=\{(0,0)\},\ B=\{(0,1)\},\ X=A\cup B\cup\{(\frac1n,y):n\in\mathbb N,0\le y\le1\}.$

Then $A$ and $B$ are disjoint components of $X,$ but there is no relatively clopen subset of $X$ which separates $A$ and $B.$