Let $(X,\tau)$ be a [topological space](/page/Topological%20Space). For $x \in X$, define the [connected component](/page/Connected%20Component) of $x$ by
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\begin{align*}
C_x := \bigcup \{A \subset X : x \in A \text{ and } A \text{ is connected in the subspace topology}\}.
\end{align*}
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Define a relation $\sim$ on $X$ by declaring, for $x,y \in X$,
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\begin{align*}
x \sim y \iff \text{there exists a connected subset } A \subset X \text{ such that } x \in A \text{ and } y \in A.
\end{align*}
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Then $\sim$ is an [equivalence relation](/page/Equivalence%20Relation) on $X$, and for every $x \in X$, the equivalence class of $x$ is exactly $C_x$: