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A two-point set must be zero-dimensional

John Kulesza
1992 Proceedings of the American Mathematical Society  
We prove that any subset of M2 that intersects no line in more than two points either is zero-dimensional or contains an arc. This is used to prove that any subset of R2 that intersects each line in exactly two points is zero-dimensional, answering a question of Mauldin.
doi:10.1090/s0002-9939-1992-1093599-1 fatcat:l6ql5za5fvbtbdv4ilusbwg4ke