Definition:Component (Topology)

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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 $\operatorname{Comp}_x \left({T}\right)$.


Definition 2

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 $\left.\right\}$


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.


Also see