I'm trying to understand the difference between components and quasicomponents. I'm using the following definitions:
$x\sim y$ iff $x$ and $y$ lie together in some connected set. The component of $x$ is the equivalence class of $x$, $C_x:=[x]_{\sim}$
$x\approx y$ iff there is no decomposition into disjoint open sets, one containing $x$ and other containing $y$. The quasicomponent of $x$ is the equivalence class of $x$, $QC_x:=[x]_{\approx}$
I can see that the component is the biggest connected set that contains $x$, and that it is a closed set, but I'm not sure what quasicomponents are. I was thinking that perhaps the definition could be seen as "$x\approx y$ iff $x,y\in U,\;\forall U\in\tau$ ", that way is clear that in general $C_x\not=QC_x$, and that $QC_x$ are open and $C_x\subset QC_x$, although that last one might not be as clear.
Am I right? Where can I find examples of this?