If $A$ dense $A$ is not connected

Question

Let $A\subseteq \mathbb{R}$ be dense in $\mathbb{R}$ but $A\neq\mathbb{R}$, Prove: $A$ is not connected

a. can $A\neq \mathbb{R}$? by definition $A$ dense iff $\overline{A}=\mathbb{R}$

b. assume that it is possible, to show that $A$ is not connected I can find an set within $A$ which is open and close? is it sufficient?

Answer 1

$A$ is not equal to $\mathbb{R}$ so pick $p \notin A$. Then $U = A \cap \{x: x > p\}$ and $a \cap \{x : x < p\}$ are non-empty (a mild use of $A$ being dense in $\mathbb{R}$), relatively open in $A$ (clear) and their union is $A$, so $A$ is disconnected.