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, if and only if 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 $\map {\operatorname {QC}_x} T$.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness