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HallsofIvy

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## Main Question or Discussion Point

Since things are a bit quiet here, I thought I would throw out a puzzle I came up with several years ago, after reading an article on connected sets:

Find two sets, P and Q, satisfying:

1) Both P and Q are completely contained in the (closed) rectangle in R

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

3) P and Q are both connected sets.

4) P and Q are disjoint.

The solution involves the difference between "connected" and "path-wise connected".

Find two sets, P and Q, satisfying:

1) Both P and Q are completely contained in the (closed) rectangle in R

^{2}with vertices at (1, 1), (1, -1), (-1, -1), and (-1, 1).2) P contains the diametrically opposite points (1, 1) and (-1, -1) while Q contains(1, -1) and (-1, 1).

3) P and Q are both connected sets.

4) P and Q are disjoint.

The solution involves the difference between "connected" and "path-wise connected".