Ultra--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 , d K ) is precompact and ( X , d X ) 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 ) ) = d X ( x , y ) and D ( ( u , x ) , ( v , x ) ) = d K ( 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.