\begin{definition}[Definition:Quasicomponent]
Let $T = \struct {S, \tau}$ be a topological space.
Let the relation $\sim$ be defined on $T$ as follows:
:$x \sim y \iff T$ is connected between the two points $x$ and $y$
That is, {{iff}} each separation of $T$ includes a single open set $U \in \tau$ which contains both $x$ and $y$.
We have that $\sim$ is an equivalence relation, so from the Fundamental Theorem on Equivalence Relations, the points in $T$ can be partitioned into equivalence classes.
These equivalence classes are called the '''quasicomponents''' of $T$.
If $x \in S$, then the '''quasicomponent of $T$ containing $x$''' (that is, the set of points $y \in S$ with $x \sim y$) can be denoted by $\map {\operatorname {QC}_x} T$.
\end{definition}