Are points in different connected components separated by open subsets? Decide if the following statement is true or false:

If $a,b \in M$ belong to different connected components, then there exists a disconnection $M = A \cup B$ (with $A$, $B$ open and disjoint), with $a \in A$ and $b \in B$.

(Hint: consider $a = (0,0)$, $b = (0,1)$ , $X = \{(1/n,y) \in \mathbb{R}^2, n \in \mathbb{N}, y \in \mathbb{R} \}$ 
and $M = X \cup \{a,b\}$ )"
I've trying to do it, but I can't prove that $M$ is a counterexample for the statement above. Could you help me?

You have to show that any open and closed (clopen) subset $A$ of $M$ which contains $a$ also contains $b$.
Hint: If $A$ is a clopen set around $a$, then it intersects all but finitely many of the intervals $J_n=\{1/n\}\times\Bbb R$. How "much" of $J_n$ would $A$ then include. Can you construct a sequence $x_n\to b$ within $A$?
