Connected Sets

[HallsofIvy]

Here's a problem I came up with many years ago:

Give two sets P and Q such that
1) P and Q are completely contained in the square $\displaystyle -1\le x\le 1$, $\displaystyle -1\le y\le 1$.
2) P contains the diagonally opposite points (-1, -1) and (1, 1).
3) Q contains the diagonally opposite points (-1, 1) and (1, -1).
4) P and Q are both connected sets.
5) $\displaystyle P\cap Q$ is empty.