Here's a puzzle I mentioned in another thread but it might be of interest here. I came up with it many years ago based on a math article I had read.

Find two sets, P and Q satisfying the following:

1) They are both subsets of the square in $\displaystyle R^2$ defined by

$\displaystyle -1\le x\le 1$ and $\displaystyle -1\le y\le 1$

2) P contains the diagonally opposite points (1, 1) and (-1, -1) while Q contains the diagonally opposite points (-1, 1) and (1, -1).

3) P and Q are both connected sets.

4) P and Q are disjoint.

The key is that while P and Q are required to be connected, they cannot be path-connected.