Line-closed subsets of Steiner triple systems and classical linear spaces

A proper non-empty subset C of the points of a linear space = (P,L) is called line-closed if any two intersecting lines of , each meeting C at least twice, have their intersection in C. We show that when every line has k points and every point lies on r lines the maximum size for such sets is r + k − 2. In addition it is shown that this cannot happen for projective spaces PG(n,q) unless q = 2, nor can it be obtained for affine spaces AG(n,q) unless n = 2 and q = 3. However, for all odd values of r there exist Steiner triple systems having such maximum line-closed subsets