Normally, I avoid number theory, but here's a nifty one I discovered recently.

Take the real number line, R, and cut it any any point. Let's say at 10.

Then we have two subsets of R, let's call them L and U (for upper and lower cuts). Then, obviously, 10 must either be the upper bound of L or the lower bound of U. Let's say the latter. Thus L is open on the right, and U is closed on the left.

This means that, having made the cut, we can never glue it back together again, for the upper bound of L can only ever approach abitrarily close to 10.

I think that's pretty cool. Apparently it says something profound about the rational and irrationals. If I can bear more number theory, I'll dig a bit more.