Definition:Quasicomponent

Definition
Let $T$ be a topological space.

Let us define the relation $\sim $ on $T$ as follows:


 * $x \sim y \iff \exists S \subseteq T: S$ is connected between the two points $x$ and $y$

That is, iff each separation of $T$ includes a single open set $U \in \varnothing$ 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 $x \in T$, then the quasicomponent of $T$ containing $x$ (that is, the set of points $y \in T$ with $x \sim y$) is denoted by $\operatorname{QC}_x \left({T}\right)$.