Definition of Connected Set in $\mathbb{R}^n$

While studying mathematical analysis, the textbook gives an definition as following:

Definition 1: A set $E\subset\mathbb{R}^n$ is a connected set that cannot be represented as the union of two disjoint nonempty sets $A,B$ which satisfy $A'\cap B=\phi$ and $A\cap B'=\phi$.

My teacher gives the following definition:

Definition 2: A set $E\subset\mathbb{R}^n$ is a connected set that cannot be represented as a subset of the union of two disjoint open sets $A,B$ which satisfy $A\cap E\neq\phi$ and $B\cap E\neq\phi$.

I try to prove they are equivalent. 1$\Rightarrow$2 is easy, but I find difficulty proving 2$\Rightarrow$1. Is it correct or not? (Sorry for my poor English)

• What are $A', B'$? – D_S Oct 4 '17 at 1:37
• the derived set of $A,B$, all limit points of $A,B$ – AbnerYe Oct 4 '17 at 1:39

They are equivalent. Suppose $A$ and $B$ witness that $E$ is not connected according to definition 1. Let $U=\{x\in\mathbb{R}^n:d(x,A)<d(x,B)\}$ and $V=\{x\in\mathbb{R}^n:d(x,A)>d(x,B)\}$. Here $d(x,A)$ means $\inf \{d(x,y):y\in A\}$. Then $U$ and $V$ are open (exercise), and they are obviously disjoint. Since $A\cap\overline{B}=\emptyset$, $d(x,B)>0$ for any $x\in A$ and so $A\subseteq U$, and similarly $B\subseteq V$. Thus $U$ and $V$ witness that $E$ is not connected according to definition 2.