Equivalence of Definitions of Component

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

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

(a) $$C$$ is the union of all connected subsets of $$T$$ that contain $$x$$.

(b) $$C$$ 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 (a).

To prove (b), note first that $$C$$ is a connected subset of $$T$$ by (a) 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 (a) we have $$A\subset C$$. This completes the proof.