If S is a set of points in space, define its line-closure L(S) = the union of all lines passing through two distinct points of S. That is: Point X lies in L(S) if there exist distinct points A, B ∈ S such that A, B, X are collinear. Then S ⊆ L(S), provided S contains at least two points. For example, if points A, B, C do not lie in a line, then L({A, B, C}) is the union of three lines whose intersection points are A, B, C. In this case, L(L({A, B, C})) is the whole plane containing those points.

1. When can a set S equal its own line-closure: L(S) = S ? Prove that your answer is correct.

2. Suppose A, B, C, D are points in 3-space that do not all lie in one plane. Describe the sets L({A, B, C, D}) and L(L({A, B, C, D})).

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