\(n\)-point sets and graphs of functions (Q946605)

An \(n\)-point set (a plane set which hits each line in exactly \(n\) points) and the graph of a function from \(\mathbb{R}\) to \(\mathbb{R}\) (a plane set which hits each vertical line in exactly one point) have simple and somewhat similar definitions, but clearly no set can be both. The question considered here is whether an \(n\)-point set can be homeomorphic to such a graph. It is known that each 2-point set with two points removed is homeomorphic to such a graph, while there exists a 2-point set which is not. The main result in this paper is that for each \(n\geq 2\), there exists an \(n\)-point set which is homeomorphic to the graph of a function from \(\mathbb{R}\) to \(\mathbb{R}\). For each \(n\geq 4\), there exist both zero-dimensional and one-dimensional examples.