Does this matroid invariant have a name?

Question

For a matroid $M$ on $X$ with closure operator $\tau:2^X\to 2^X$ let $c(M)=\min\{|S|:\tau(X\setminus S)\neq X\}$. This is an invariant because if $M$ and $M'$ are isomorphic (i.e. if flats of $M$ are bijective images of flats of $M'$) then $c(M)=c(M')$ likewise note if $M$ is the graphic matroid of any undirected graph $G$ then $c(M)=\lambda(G)$ is the edge connectivity of $G$ also in particular if $M$ is a gammoid then $c(M)$ is equal to the maximum number of pairwise disjoint bases of $M$. Anyway probably over thinking this as I'm very new to matroid theory, perhaps studying the dual of $M$ might simplify things?

Answer 1

This is the cogirth of the matroid. Note that a set $S$ satisfies $\tau (X\setminus S)\neq X$ if and only if $X\setminus S$ is not spanning. This is the same as asking that $S$ is a dependent set in $M^*$. Thus, your condition can be rewritten as $$c(M)=\min\{|S| : S\text{ is a dependent set of }M^*\}$$ which is the definition of cogirth.