Résumé : CAT(0) cube complexes are have a description in terms of median algebras. This description lends itself to generalization, one of which is the notion of a topological mdian algebra. We will discuss an initial exploration into a characterization of topological median algebra structures on euclidean spaces in terms what we term “web structures”.  This characterization works for ER homology manifolds as well. This is joint work with Bestvina and Bromberg.