Let $X$ be a compact connected Hausdorff space and $M$ be the set of compact connected subspaces of $X$.
Let $\mathscr{C}$ be a nonempty chain in $(M,\subset)$.
Let $a:\mathscr{C}\rightarrow \bigcup \mathscr{C}$ be a choice function.
How do I prove that there exists a point $x$ in $X$ such that for every neighborhood $U$ of $x$ in $X$, there exists $Y_0\in \mathscr{C}$ such that $x\in Y$ for all $Y\in \mathscr{C}$ with $Y\subset Y_0$?