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 G δ -separation property for the class G δ 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 ) .