Define the notion of a connected component of a space $X$.

If $A_{\alpha} \subset X$ are connected subsets of $X$ such that $\bigcap_{\alpha} A_{\alpha} \neq \emptyset$, show that $\bigcup_{\alpha} A_{\alpha}$ is connected.

Prove that any point $x \in X$ is contained in a unique connected component.

Let $X \subset \mathbb{R}$ consist of the points $0,1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{n}, \ldots$. What are the connected components of $X$ ?