Linear spaces with projective lines (Q1849896)

A linear space is a geometry of points and lines with the property that every two different points are contained in a unique line. For a linear space we have the well-known notions of subspace and dimension of a subspace. A plane of a linear space is a subspace of dimension 2. For every set \(U\) of points, we define \(\overline{U}\) as the smallest subspace through \(U\). A line \(L\) in a linear space is called a projective line if \(L\) intersects every line of the plane \(\overline{L \cup \{ x \}}\) for every point \(x\) not on \(L\). A linear space is called locally projective if for every point \(x\) the lines and planes through \(x\) form the points and lines of a projective space. The Bundle Theorem states that for four lines \(A\), \(B\), \(C\) and \(D\), no three in a common plane, the coplanarities of \(\{ A,B \}\), \(\{ A,C \}\), \(\{ A,D \}\), \(\{ B,C \}\), \(\{ B,D \}\) imply the coplanarity of \(\{ C,D \}\). The author considers linear spaces which satisfy the following two properties. (I) For any two planes \(E_1\) and \(E_2\) intersecting in a line \(G\), there exist distinct points \(p\) and \(q\) on \(G\) and projective lines \(K_1\), \(K_2\), \(L_1\) and \(L_2\) such that \(L_i,K_i \subset E_i\) (\(i \in \{ 1,2 \}\)), \(p=L_1 \cap L_2\) and \(q=K_1 \cap K_2\). (II) For every two intersecting lines \(H_1\) and \(H_2\) of a plane \(F\) and every point \(x\) in \(F\), there exists a line \(G\) through \(x\) such that \(\emptyset \not= G \cap H_1 \not= G \cap H_2 \not= \emptyset\). The author proves that the Bundle Theorem holds in any such linear space \(S\). He also proves that \(S\) is locally projective. By the Theorem of Kahn we then know that \(S\) is embeddable in a projective space.