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.