TITLE: Prove that each component of $X$ is a closed subset of $X$.
QUESTION [8 upvotes]: 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.

REPLY [13 votes]: I think I came up with a simpler proof: That components are the largest connected subsets of $X$; I choose one of them, say $C$. Since C is connected in X, thus so is $Cl(C)$. Thus $C=Cl(C)$ and then $C$ is closed.