Give an example of a topological space $X$, and a connected component which is not open in $X$ 
Give an example of a topological space $X$, and a connected component which is not open in $X$

I know of the following theorems:


*

*Each connected component is closed;

*If a topological space has a finite number of connected components, then those components are open in $X$.


Because of the second theorem, I need to be on the lookout for a topological space with a infinite number of connected components. I was thinking of something like this:
$$X = \bigsqcup_{n\in \mathbb{N}} \left[n, \dfrac{3}{2} n\right[$$
But this doesn't seem to work since $\sqcup_{n\geqslant 2} [n, \frac{3}{2}[$ as an union of infinite closed sets seems to be closed again. 
The open/closed concept in a relative topology can be mindbending. Could someone help me to find an simple example?

Take $$X=\mathbb{Q}$$ the rationals numbers with the Euclidean topology.
