Definition:Component (Topology)/Definition 2
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Definition
Let $T = \struct{S, \tau}$ 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:
- $\ds \map {\operatorname{Comp}_x} T = \bigcup \leftset{A \subseteq S: x \in A \land A}$ is connected $\rightset{}$