Sequence of connected closed subsets of plane? Suppose $S_1, S_2, S_3, ...$ is a sequence of connected, closed subsets of the plane and $S_1 \supset S_2 \supset ...$ Is $S = \cap S_n$ connected?
My reasoning is that S is connected because $S$ = the last $S_i$ in the sequence, as they are all nested. Since all of the $S_i$'s were defined by construction to be connected and closed subsets of the plane, S is also connected. Is this reasoning valid?
The second part of the question states: does the answer change if the sets are compact as well?
My answer: no, as this does not alter my reasoning for the first part of the question.
Am I completely off, or am I on the right track? Thanks!

Hint 1: Answer to first part of the question is not necessarily.
Come up with a counter example yourself. Think outside the box.

Hint 2: Answer to second part of the question is yes.
Now that you know the answer is yes, what does that tell you about the sets in the previous counter example?
Also, prove this!
