Equivalence of Definitions of Component

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

Let $x \in T$.

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

Let $C = \bigcup \mathcal C_x$

From Union of Connected Sets with Common Point is Connected, $C$ is a connected set of $T$.

Furthermore, $x \in C$.

Hence $C \in \mathcal C_x$.

Let $C’$ be the equivalence class containing $x$ of the equivalence relation $\sim$ defined by:
 * $y \sim z$ $y$ and $z$ are connected in $T$.

Definition 1 if and only if Definition 2
It needs to be shown that $C = C’$.

The result follows.

Definition 2 implies Definition 3
From Set is Subset of Union,
 * $\forall B \in \mathcal C_x : B \subseteq C$.

It follows that $C$ is a greatest set in $\mathcal C_x : B \subseteq C$.

From Greatest Element is Maximal, $C$ is maximal.

Definition 3 implies Definition 2
Let $\tilde C$ be a maximal connected set of $T$ that contains $x$.

By definition, $\tilde C \in \set {A \subseteq S : x \in A \land A \text{ is connected in } T}$

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

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

Also see

 * Connectedness of Points is Equivalence Relation