Equivalence of Definitions of Component

Theorem
Let $T$ be a topological space, and let $x \in T$.

Then the component $C = \operatorname{Comp}_x \left({T}\right)$ of $T$ containing $x$ can be characterized as follows:


 * $(1): \quad C$ is the union of all connected subsets of $T$ that contain $x$.
 * $(2): \quad C$ is the maximal connected subspace of $T$ that contains $x$.

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 $(1)$.

To prove $(2)$, note first that $C$ is a connected subset of $T$ by $(1)$ and the fact that Spaces with Connected Intersection have Connected Union.

On the other hand, if $A$ is a connected subset of $T$ that contains $x$, then by $(1)$ we have $A \subseteq C$. This completes the proof.