Does the limit of a descending sequence of connected sets still connected?

Solution 1:

No. Let $F_n$ be the the plane $\mathbb R^2$ minus the line $\{0\}\times(-\infty,n)$.

Added: It is true when all the $F_n$ are compact subsets of $\mathbb R^N$. Suppose otherwise: then there exist open disjoint sets $A,B$ such that $F$ contains points of both $A$ and $B$ and $F$ is contained in $A\cup B$. Now consider $F_n\cap (\partial A)$. Since each $F_n$ is connected, and contains points in both $A$ and $B$, the intersection $F_n\cap (\partial A)$ must be nonempty, and moreover, for $n=1,2,3,\ldots$ it is a decreasing sequence of compact sets, and therefore the intersection of all $F_n\cap (\partial A)$ is nonempty. Contradiction. Thus $F$ is connected.