Closure Spaces and Matroids

Abstract

Let G be a projective geometry. The associated closure operator C : P(G) → P(G) sends a subset A ⊆ G to the smallest subspace of G which contains A. Given two points a, b ∈ G one has C({a,b}) = a ⋆ b. Hence the closure operator C gives much more information than the operator ⋆ does, and it is therefore not surprising that a projective geometry can be described as a set G together with a closure operator C : P(G) → P(G) satisfying some additional axioms.