A Characterization of Projective Spaces by a Set of Planes

Abstract

This note deals with the following question: How many planes of a linear space (P,\(\mathfrak{L}\)) must be known as projective planes to ensure that (P,\(\mathfrak{L}\)) is a projective space? The following answer is given: If for any subset M of a linear space (P,\(\mathfrak{L}\)) the restriction (M,\(\mathfrak{L}\))(M)) is locally complete, and if for every plane E of (M,\(\mathfrak{L}\)(M)) the plane \(\bar E\) generated by E is a projective plane, then (P,\(\mathfrak{L}\)) is a projective space (cf. 5.6). Or more generally: If for any subset M of P the restriction (M,\(\mathfrak{L}\)(M)) is locally complete, and if for any two distinct coplanar lines G1, G2 ∈ \(\mathfrak{L}\) (M) the lines \(\bar G_1 ,\bar G_2 \varepsilon \mathfrak{L}\) generated by G1, G2 have a nonempty intersection and \(\overline {G_1 \cup {\text{ }}G_2 }\) satisfies the exchange condition, then (P,\(\mathfrak{L}\)) is a generalized projective space.