Definition:Quasicomponent

Definition
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, 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 $\map {\operatorname {QC}_x} T$.