Definition:Component (Topology)/Definition 2

Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let the relation $\sim$ be defined on $T$ as follows:

$x \sim y$ if and only if $x$ and $y$ are connected in $T$.

That is, if and only if there exists a connected set of $T$ that contains both $x$ and $y$.

The component of $T$ containing $x$ is defined as:

$\displaystyle \operatorname{Comp}_x \left({T}\right) = \bigcup \left\{{A \subseteq S: x \in A \land A}\right.$ is connected $\left.\right\}$