Let X be a topological space.
Given a decreasing sequence of closed and connected subsets $J_1\supset J_2 \supset J_3\supset\cdots$, consider $J:=\bigcap_{i=1}^\infty J_i$.
Is it always true that $J$ is connected when $X=\mathbb{R}^2$ ?
and when $X$ is compact but not Hausdorff ? (if $X$ is compact and Hausdorff $J$ is connected)
Best Answer
Counterexample:
Consider $J_n=(\mathbb R^\times\times \mathbb R)\cup (\{0\}\times (n,\infty))$. Each $J_n$ is connected, but the intersection is $\mathbb R^\times\times\mathbb R$ which is disconnected. Basically, each $J_n$ eats away a bit more of the connection, and eventually every part of the connection gets consumed.
Edit: An edit to the question now added the condition of closed sets, which is violated by the example above. But the basic idea can still be used.
Now consider $J_n = (((-\infty,-1]\cup[1,\infty))\times\mathbb R)\cup ([-1,1]\times [n,\infty))$. It is not hard to check that those sets are closed and connected, but the intersection is $((-\infty,-1]\cup[1,\infty))\times\mathbb R$ which is disconnected.