I know that in a compact Hausdorff space the quasi-components of the points coincide with their connected components, but I'd like to know if the Hausdorff hypothesis is needed.
I could not find a counterexample, i.e. a compact space in which a connected component is strictly contained in a quasi-component. I tried with the finite-complement topology, but it is connected. So it cannot be a counterexample because such a space must have infinite connected components.
So my question is: is there such a counterexample?
P.S. Sorry for my bad English, but I'm Italian and don't speak English very well.
Here's a simple example. Let $X=\mathbb{N}\cup\{a,b\}$, where a set $U\subseteq X$ is open iff $U\subseteq\mathbb{N}$ or $U$ is cofinite. The clopen subsets of $X$ are the finite subsets of $\mathbb{N}$ and the cofinite sets which contain both $a$ and $b$. In particular, $X$ is totally disconnected and compact, but $a$ and $b$ are in the same quasicomponent.