Finding connected components of a topology space I would like to understand how to find the connected components of a given topological space.
I have this example, that I understand:
If I have $\mathbb{Q}$ with the euclidean topology, the connected components are just the singleton elements of $\mathbb{Q}$.
Proof: If I want to show that the only connected sets are the singletons, I would take a set that is not a singleton and I call it $C$. There exist $x,y\in C$ such that $x \neq y$. Then take $(-\infty,i)$ and $(i,\infty)$ as a seperation for $C$ ($i$ being the irrational).
But if I have for example $\mathbb{Q}$ with another topology i.e cofinite or conumerable. Or just $\mathbb{R}$ with cofinite or conumerable how should i proceed?

You have to study case by case. For example cofinite topology on an infinite set is connected simply because there are no disjoint open subsets in it. As you can see this is a very different approach from the one you used for the standard $\mathbb{Q}$.
There is no general method, unfortunately. There are some tools (e.g. continuous image of a connected space is connected), but ultimately it boils down to some level of cleverness.
