Connectedness of each set in a countable family of sets
Counter-examples. Let $X=\Bbb R.$
(1). $A=\{(-\infty, n)\cup (n+1,\infty):n\in\Bbb Z\}.$
(2). $A=\{(-\infty, n)\cup (n+1,\infty)\cup (0,1):n\in\Bbb Z\}.$
In both cases, no member of $A$ is connected and $\cup A=\Bbb R.$ In (1) we have $\cap A=\emptyset$. In (2) we have $\cap A=(0,1).$
To confirm this, consider first $x=n\in \Bbb Z,$ and second, $x\in (n,n+1)$ with $0\ne n\in\Bbb Z,$ and third, $x\in (0,1).$