(a) Let $X$ be a topological space and suppose $X=C \cup D$, where $C$ and $D$ are disjoint nonempty open subsets of $X$. Show that, if $Y$ is a connected subset of $X$, then $Y$ is entirely contained in either $C$ or $D$.

(b) Let $X$ be a topological space and let $\left\{A_{n}\right\}$ be a sequence of connected subsets of $X$ such that $A_{n} \cap A_{n+1} \neq \emptyset$, for $n=1,2,3, \ldots$. Show that $\bigcup_{n \geqslant 1} A_{n}$ is connected.