Connected Components Partition the Space
# Statement
Suppose $X$ is a Topological Space. Then the collection $$\mathcal{C} := {C \subset X : C \text{ is a nonempty connected component}}$$ is a Partition of $X$.
# Proof
We must show
- $\emptyset \not\in \mathcal{C}$
- $C \cap C’ \neq \emptyset \Rightarrow C = C’$ for $C, C’ \in \mathcal{C}$
- $\bigcup\limits_{C \in \mathcal{C}} C = X$
(1) follows from our construction of $\mathcal{C}$. For (2), note that if $C \cap C’ \neq \emptyset$, then by If Connected Sets share a point, then their Union is Connected, $C \cup C’$ is Connected. $C \cup C’ \supset C, C’$ so $C = C \cup C’ = C’$ because both $C$ and $C’$ are Connected Components.
For (3), let $x \in X$ be arbitrary. Then consider $\mathcal{C}{x} = {D \subset X : x \in D, D \text{ is connected}}$. By If Connected Sets share a point, then their Union is Connected, $C{x} := \bigcup\limits_{D \in \mathcal{C}{x}} D$ is Connected. If there were some Connected $E \supset C{x}$, then $x \in E$ so $E \in \mathcal{C}{x}$ making $E \subset C{x}$, so $E = C_{x}$. Thus $C_{x}$ is a Connected Component containing $x \in X$. Thus taking $\bigcup\limits_{x \in X} C_{x} = X$ shows (3) and completes the proof $\blacksquare$.