Let X be a topological space. Define a binary relation $\sim$ in $X$ as follows: $x \sim y$ if there exists a connected subspace $C$ included in $X$ such that $x,y$ belong to $C$. Show the following.
(i) $\sim$ is an equivalence relation.
(ii) Each equivalence class is a maximal connected subspace of $X$. These equivalence classes are called the connected components of $X$.
(iii) Each connected component is a closed subset of $X$. To this end, show that the closure of a connected set is connected.