Definition:Component (Topology)/Definition 2

Definition
Let $T = \struct{S, \tau}$ be a topological space.

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


 * $x \sim y$ $x$ and $y$ are connected in $T$.

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

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


 * $\ds \map {\operatorname{Comp}_x} T = \bigcup \leftset{A \subseteq S: x \in A \land A}$ is connected $\rightset{}$

Also see

 * Equivalence of Definitions of Component