I need some help on following question.
Let be (X, $\Omega_x$) a topological space.
Suppose that or for every $n\in\mathbb N$, $F_n$ is a closed and connected set, such that for every n < m, $F_m \subseteq F_n$
Proof or disproof: the set $\bigcap_{n = 1}^\infty F_n $ is also connected.
I think that if (X, $\Omega_x$) is a metrizable space, then the above claim is true, but not for general topological space.
Unfortunately, I don't succeed to find a counterexample..
Any help will be appreciated!!
Start with $\mathbb R^2$ and remove the line segment $(-1,1)\times \{0\}$. Call the resulting space $X$. Let $A=[1,\infty)\times \mathbb R$ and $B=(-\infty,-1]\times \mathbb R$ and set $C_n =( [-1,1]\times [-\frac{1}{n},\frac{1}{n}])\cap X$. Then $F_n = A\cup B\cup C_n$ should provide a counter example, since $\bigcap F_n = A\cup B$ in $X$, which is not connected.