Logic Colloquium 2004, Tutorial on Geometric Stability Theory, Combinatorial Geometry

Combinatorial geometry

Definition: A combinatorial pregeometry is a set X given together with a function (called the closure operation) cl:P(X) → P(X) on the power set of X satisfying

cl(cl(A)) = cl(A)
A ⊆ cl(A)
A ⊆ B ⇒ cl(A) ⊆ cl(B)
cl(A) = ∪ { cl(F) | F ⊆ A finite}
a ∈ cl(A ∪ { b }) - cl(A) ⇒ b ∈ cl(A ∪ { a })

A combinatorial geometry is a pregeometry satisfying in addition

cl(∅) = ∅
cl({a}) = {a}

Proposition/Definition: If (X,cl) is a combinatorial pregeometry, then we say that a subset A ⊆ X is independent if a ∉ cl(A - { a}) for every a ∈ A. If A, B ⊆ X are two maximal independent subsets of X, then A and B have the same cardinality. By definition, the dimension of X, dim(X), is the cardinality of a maximal independent subset.