Equivalence of Definitions of Component

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Theorem

The following definitions of the concept of Component in the context of Topology are equivalent:


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

Let $x \in T$.

Definition 1: Equivalence Class

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: Union of Connected Sets

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: Maximal Connected Set

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

the maximal connected set of $T$ that contains $x$.


Proof

Let $\mathcal C_x = \set {A \subseteq S : x \in A \land A \text{ is connected in } T}$

Let $C = \bigcup \mathcal C_x$

Lemma

$C$ is connected in $T$ and $C \in \mathcal C_x$.


Let $C’$ be the equivalence class containing $x$ of the equivalence relation $\sim$ defined by:

$y \sim z$ if and only if $y$ and $z$ are connected in $T$.


Equivalence Class equals Union of Connected Sets

It needs to be shown that $C = C’$.

\(\displaystyle y \in C’\) \(\leadstoandfrom\) \(\displaystyle \exists B \text{ a connected set of } T, x \in B, y \in B\) Definition of $\sim$
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle \exists B \in \mathcal C_x : y \in B\) Equivalent definition
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle y \in \bigcup \mathcal C_x\) Definition of Set Union
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle y \in C\) Definition of $C$

The result follows.

$\Box$

Union of Connected Sets is Maximal Connected Set

Let $\tilde C$ be any connected set such that:

$C \subseteq \tilde C$

Then $x \in \tilde C$.

Hence $\tilde C \in \mathcal C_x$.

From Set is Subset of Union,

$\tilde C \subseteq C$.

Hence $\tilde C = C$.

It follows that $C$ is a maximal connected set of $T$ by definition.

$\Box$

Maximal Connected Set is Union of Connected Sets

Let $\tilde C$ be a maximal connected set of $T$ that contains $x$.

By definition, $\tilde C \in \mathcal C_x$.

From Set is Subset of Union, $\tilde C \subseteq C$.

By maximality of $\tilde C$ then $\tilde C = C$

$\blacksquare$


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