Equivalence of Definitions of Component

From ProofWiki
Jump to navigation Jump to search

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 $\map {\operatorname {Comp}_x} T$.

Definition 2: Union of Connected Sets

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: Maximal Connected Set

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

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


Proof

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

Let $C = \bigcup \CC_x$


Lemma

$C$ is connected in $T$ and $C \in \CC_x$.

$\Box$


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 $C = C'$.

\(\ds y \in C'\) \(\leadstoandfrom\) \(\ds \exists B \text{ a connected set of } T, x \in B, y \in B\) Definition of $\sim$
\(\ds \) \(\leadstoandfrom\) \(\ds \exists B \in \CC_x : y \in B\) equivalent definition
\(\ds \) \(\leadstoandfrom\) \(\ds y \in \bigcup \CC_x\) Definition of Union of Set of Sets
\(\ds \) \(\leadstoandfrom\) \(\ds y \in C\) 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 $\tilde C \in \CC_x$.

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:

$\tilde C \in \CC_x$

From Set is Subset of Union:

$\tilde C \subseteq C$

By maximality of $\tilde C$:

$\tilde C = C$

$\blacksquare$


Also see


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Equivalence_of_Definitions_of_Component&oldid=469581"