  Given a finite set $E$ and an operator $\sigma:2^{E}\longrightarrow2^{E}$,
two sets $X,Y\subseteq E$ are \textit{cospanning} if $\sigma\left( X\right)
=\sigma\left( Y\right) $. Corresponding \textit{cospanning equivalence
relations} were investigated for greedoids in much detail (Korte, Lovasz,
Schrader; 1991). For instance, these relations determine greedoids uniquely. In
fact, the feasible sets of a greedoid are exactly the inclusion-wise minimal
sets of the equivalence classes. In this research, we show that feasible sets
of convex geometries are the inclusion-wise maximal sets of the equivalence
classes of the corresponding closure operator. Same as greedoids, convex
geometries are uniquely defined by the corresponding cospanning relations. For
each closure operator $\sigma$, an element $x\in X$ is \textit{an extreme
point} of $X$ if $x\notin\sigma(X-x)$. The set of extreme points of $X$ is
denoted by $ex(X)$. We prove, that if $\sigma$ has the anti-exchange property,
then for every set $X$ its equivalence class $[X]_{\sigma}$ is the interval
$[ex(X),\sigma(X)]$. It results in the one-to-one correspondence between the
cospanning partitions of an antimatroid and its complementary convex geometry.
The obtained results are based on the connection between violator spaces,
greedoids, and antimatroids. Cospanning characterization of these combinatorial
structures allows us not only to give the new characterization of antimatroids
and convex geometries but also to obtain the new properties of closure
operators, extreme point operators, and their interconnections.

Combinatorics;Discrete Mathematics