Equivalence of Definitions of Path Component

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Theorem

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


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

Let $x \in T$.


Equivalence Class

Let $\sim$ be the equivalence relation on $T$ defined as:

$x \sim y \iff x$ and $y$ are path-connected.

The equivalence classes of $\sim$ are called the path components of $T$.

If $x \in T$, then the path component of $T$ containing $x$ (that is, the set of points $y \in T$ with $x \sim y$) can be denoted by $\map {\operatorname{PC}_x} T$.

Union of Path-Connected Sets

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

$\displaystyle \map {\operatorname{PC}_x} T = \bigcup \left\{{A \subseteq S: x \in A \land A}\right.$ is path-connected $\left.\right\}$

Maximal Path-Connected Set

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

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


Proof

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

Let $C = \bigcup \CC_x$.


Lemma

$C$ is path-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 Path-Connected Sets

It needs to be shown that $C = C'$.

\(\displaystyle y \in C'\) \(\leadstoandfrom\) \(\displaystyle x \text{ is path-connected to } y \text{ in } T\) Definition of $\sim$
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle \exists B \text{ a connected set of } T, x \in B, y \in B\) Points are Path-Connected iff Contained in Path-Connected Set
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle \exists B \in \CC_x : y \in B\) Equivalent definition
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle y \in \bigcup \CC_x\) Definition of Set Union
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle y \in C\) Definition of $C$

The result follows.

$\Box$


Union of Path-Connected Sets is Maximal Path-Connected Set

Let $\tilde C$ be any path-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 path-connected set of $T$ by definition.

$\Box$


Maximal Path-Connected Set is Union of Path-Connected Sets

Let $\tilde C$ be a maximal path-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 $\tilde C$:

$\tilde C = C$

$\blacksquare$


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