Equivalence of Definitions of Component

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $x \in T$.

Proof
By definition, $y \in C$ if and only if there exists a connected subset of $T$ that contains both $x$ and $y$. This proves $(1) \iff (2)$.

To prove $(2) \iff (3)$, note first that $C$ is a connected subset of $T$ by $(2)$ and Space with Connected Intersection has Connected Union.

On the other hand, if $A$ is a connected subset of $T$ that contains $x$, then by $(2)$ we have $A \subseteq C$. This completes the proof.