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) $$C$$ is the union of all connected subsets of $$T$$ that contain $$x$$.
 * 2) $$C$$ 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.