Let $X$ be a topological space and let $\left\{ Y_{i}\right\} _{i=1}^{\infty}$
be a descending chain of closed connected subsets of $X$. I know from reading elsewhere that ${\displaystyle \bigcap_{i=1}^{\infty}Y_{i}}$ is not necessarily a connected subspace of $X$ but I have no counter example and I haven't managed to come up with one.
There is a counter example here to the same question while also assuming $X$ is compact. However, it uses the quotient topology which I haven't studied about so I would prefer a different counterexample that does not use the quotient topology.
Help would be appreciated!
Best Answer
In $\mathbb{R}^{2}$ take two disjoint infinite lines $L_{1},L_{2}$ connected by vertical lines $B_{1},...$ (like a ladder). Then define $$A_{n}=L_{1}\cup L_{2}\cup \bigcup_{k\geq n}B_{k}.$$
These form a descending chain and their intersection will be $L_{1}\cup L_{2}$.