If $A$ is disconnected set, then can it be written as union of connected separated sets?

Question

I wonder if this statement

If disconnected set $A$ is union of two separated sets $B$, $C$, then $B$ $C$ are connected

is true or false? I try to construct counterexample on $\mathbb{R}$ but always have $B,C$ be connected.

Editted: I think I didn't state correctly in the first time, apologize for that

If $A$ is disconnected set, then it can always be written as union of connected separated sets

Answer 1

Try taking $A = (0,1) \cup (2,3) \cup (3,4) \cup (4,5)$?

For your edit: the comment below your question is helpful

Answer 2

For every topological space $(A,\tau)$, one-point sets $\{ x\}$ must be connected, once the definition of disconnected spaces implies they must have at least two disjoint open subsets- which is clearly not true for $\{ x\}$. Therefore, if we define $A=\{ x_1,x_2,...\}=\{ x_1\} \cup\{ x_2\} \cup...$, it is clear that $\{ x_1\},\{ x_2\} ,...$ are all connected subsets. As a consequence, $A$ is the union of connected spaces, as we wanted to prove. After taking this into consideration, it would be interesting to ask what would be the "minimal" representation of $A$ as an union of connected subsets. As a matter of fact, this depends mostly on the topology $\tau$ given to $A$. These "maximal connected islands" which would be part of that minimal representation are called connected components, and are a well studied subject in general topology.

Hope I've helped in something :)