Equivalence of Definitions of Path Component
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:
- $\ds \map {\operatorname{PC}_x} T = \bigcup \leftset {A \subseteq S: x \in A \land A}$ is path-connected $\rightset {}$
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 $C = C'$.
\(\ds y \in C'\) | \(\leadstoandfrom\) | \(\ds x \text{ is path-connected to } y \text{ in } T\) | Definition of $\sim$ | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \exists B \text{ a connected set of } T, x \in B, y \in B\) | Points are Path-Connected iff Contained in Path-Connected Set | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \exists B \in \CC_x : y \in B\) | 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 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 $\tilde C$:
- $\tilde C = C$
$\blacksquare$
Also see
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness