A Characterization of Projective Spaces by a Set of Planes

This note deals with the following question: How many planes of a linear space (P, ℒ) must be known as projective planes to ensure that (P, ℒ) is a projective space? The following answer is given: If for any subset M of a linear space (P, ℒ) the restriction (M, ℒ(M)) is locally complete, and if for every plane E of (M, ℒ(M)) the plane Ē generated by E is a projective plane, then (P, ℒ) is a projective space. Or more generally: If for any subset M of P the restriction (M, ℒ(M)) is locally complete, and if for any two distinct coplanar lines G₁, G₂ ∈ ℒ(M) the lines Ḡ₁, Ḡ₂ ∈ ℒ generated by G₁, G₂ have a nonempty intersection and G₁ ∪ G₂ satisfies the exchange condition, then (P, ℒ) is a generalized projective space.