Definition 3.2.4 (Connected Component).label Let $X$ be a topological space then the (connected) components of $X$ are the equivalence classes of the equivalence relation
\begin{align*}x\sim y\iff \exists C\suf X\text{ connected subspace such that}, x, y\in C\end{align*}
Proof. By Proposition 3.2.3(2) this is an equivalence relation and since the connected component containing $X$ is the largest connected subset of $X$ containing $x$ it is closed by Proposition 3.2.3(1).$\square$
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