Geometrical mappings of Grassmannians of linear spaces satisfying the exchange axiom

Summary.

Let Π and Π′ be linear spaces satisfying the exchange axiom and having the same dimension. Denote by ${\user1{{\mathcal{G}}}}_{k} {\left( \Pi \right)}$ and ${\user1{{\mathcal{G}}}}_{k} {\left( {\Pi '} \right)}$ the Grassmannians of k-dimensional subspaces of Π and Π′, respectively. We study mappings of ${\user1{{\mathcal{G}}}}_{k} {\left( \Pi \right)}$ to ${\user1{{\mathcal{G}}}}_{k} {\left( {\Pi '} \right)}$ which send base subsets to base subsets.