Definition: Let $X$ be a topological space and let $\sim_C$ be the equivalence relation on $X$ defined by $x \sim_C y$ if $x$ and $y$ lie in a connected subset of $X$. The components of $X$ are the equivalence classes of the equivalence relation $\sim_C$.
Question: Prove that each component of $X$ is a closed subset of $X$.
Please guide me how to start to prove, or any clue. Thank you.