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.
Best Answer
Hint: (i) I guess you're ok with $x \sim x$ and $x\sim y \Rightarrow y \sim x$. For transitivity, recall that the union of two connected sets with nonempty intersection is also a connected set.
(ii) Use the same fact of (i) (possibly with infinite elements) to check that the equivalence classes are connected. If C is a connected set in $X$, note that any two points in $C$ are equivalent, so they all must be contained in an equivalence class.
(iii) Closure of a connected subset of $\mathbb{R}$ is connected?