# 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.