Definition:Component (Topology)
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 $y$.
Definition 1
From Connectedness of Points is Equivalence Relation, $\sim$ is an equivalence relation.
From the Fundamental Theorem on Equivalence Relations, the points in $T$ can be partitioned into equivalence classes.
These equivalence classes are called the (connected) components of $T$.
If $x \in S$, then the component of $T$ containing $x$ (that is, the set of points $y \in S$ with $x \sim y$) is denoted by $\map {\operatorname {Comp}_x} T$.
Definition 2
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 $\rightset{}$
Definition 3
The component of $T$ containing $x$ is defined as:
- the maximal connected set of $T$ that contains $x$.
Also known as
A component of $T$ is also known as a connected component.
For simplicity of presentation, $\mathsf{Pr} \infty \mathsf{fWiki}$ takes the position that a component is a connected set by definition, and so it is unnecessary and unwieldy to include the word connected when using it.