A topological median structure on a topological space X is acontinuous map m:X^3->X satisfying certain axioms. A CAT(0)cube complex X has a natural median structure, where m(a,b,c) is theunique point that belongs to three l^1-gedesics that connect each pairab, ac, bc. In a work in progress, joint with Ken Bromberg and MichahSageev, we show that median structures on R^n are locally induced bycubulations of neighborhoods. The proof is by induction on n, andrequires us to prove the same theorem for ER homology manifolds.