# Definition:Quasicomponent

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## Definition

Let $T = \left({S, \tau}\right)$ 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 $\operatorname{QC}_x \left({T}\right)$.

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 4$