Union of two 'not-connected' sets of real numbers

Question

If $P$ and $Q$ are two 'not-connected' subsets of the set of all real number then is the union of $P$ and $Q$ not connected?

I haven't found any counterexample yet but the answer is given 'no'.someone please provide a counterexample.

Answer 1

Perhaps consider the set of rational number and the set of irrational numbers.

Answer 2

How about $P$ being the set of rationals, and $Q$ the set of irrationals?

Or $P=[0,1]\cup[2,3]$ and $Q=[1,2]\cup[3,4]$?

Answer 3

Consider the following fact:

In $\mathbb{R}$, the complement of a disconnected set with at least $3$ connected components is also disconnected.

This enables you to construct many counterexamples, including every possible case with $P \cap Q = \emptyset$. Simply let $P$ be any disconnected set with at least $3$ "gaps", and let $Q = \mathbb{R} \setminus P$. These are disconnected, yet $P \cup Q = \mathbb{R}$ is connected. Of course this process can also be carried out on any connected subset of $\mathbb{R}$.