Ultra-m-separability

A metric space X is ultra-m-separable if the weight of the Katětov hull, E(X), of X is no greater than m. It is shown that the collection of all nonempty ultra-m-separable subsets of X is an ideal closed under taking the limit of its members with respect to the Hausdorff distance. As an application of this, it is proved that if (K,dK) is precompact and (X,dX) is ultra-m-separable, then (K×X,D) is ultra-m-separable as well, where D is any metric on K×X such that D((u,x),(u,y))=dX(x,y) and D((u,x),(v,x))=dK(u,v) for any u,v∈K and x,y∈X. Bounded ultra-m-separable spaces are characterized by means of their metrically discrete subsets.