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

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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 $C = \bigcup \mathcal C_x$

Lemma

$C$ is connected in $T$ and $C \in \mathcal C_x$.


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

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

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

$\blacksquare$

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