Equivalence of Definitions of Component/Maximal Connected Set is Union of Connected Sets

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

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

Then:
 * $\tilde C = \bigcup \set {A \subseteq S : x \in A \land A \text{ is connected in } T}$.

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 $C \in \mathcal C_x$.

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

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

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

Also see

 * Connectedness of Points is Equivalence Relation