Separable reduction of local metric regularity

We prove that the property of a set-valued mapping F : X ⇉ Y F:X \rightrightarrows Y to be locally metrically regular (and consequently, the properties of the mapping to be linearly open or pseudo-Lipschitz) is separably reducible by rich families of separable subspaces of X × Y X\times Y . In fact, we prove that, moreover, this extends to computation of the functor {reg} F \textrm {{reg}}\, F that associates with F F the rates of local metric regularity of F F near points of its graph.