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Earlier quoted context omitted.
This is true for finite sets. For infinite sets, the Sierpinski triangle is a counterexample.
That’s an uncountable set. If we want a counter example for uncountable sets a simpler example is a circular area. Anyone happen to know if it is true for countably infinite sets?
That's countable you just go in a spiral.
I might be too stupid to understand why this is interesting and useful. If it helps I am a working physicist, and a lot of pure math is lost on me. I think I followed this, but I don't know why one would care or this would be interesting.
It is interesting to a pure mathematician. Since it is obviously true, one's intuition is that it should have a simple proof. In particular, the obvious induction ought to work. The base case is n=2. The line joining them passes through exactly two points because that is all you have. Now we attempt the induction step. We have n+1 points. Leave one, p, out. We know that the theorem applies to the n points by the indu…