Separation properties between the σ-compactness and Hurewicz property

We introduce and study so-called C-separation properties leading to a fine hierarchy of spaces with the Hurewicz property ⋃fin(O,Γ). By definition, a topological space X has the C-separation property for a class C of spaces if for any embedding X⊂C into a space C∈C there is a σ-compact subset A⊂C containing X. It turns out that the classical Hurewicz property is equivalent to the Gδ-separation property for the class Gδ of Polish spaces. On the other extreme there are Sierpiński sets having the UM-separation property for the class UM of universally measurable spaces. We construct several examples distinguishing the C-separation properties for various descriptive classes C and also study the interplay between the C-separation properties and the selection principles ⋃fin(C,Γ).