LAUSR

Abstract For a given space X let L ( X ) be the family of all compact subsets of X. A space X is dominated by the space M if X has an M-ordered compact cover, which means that there exists a family F = { F K : K ∈ L ( M ) } ⊂ L ( X ) such that ⋃ F = X and K ⊂ L implies that F K ⊂ F L , whenever K , L ∈ L ( M ) . A space X is strongly dominated by a space M if it has an M-ordered compact cover F such that for any compact K ⊂ X there is F ∈ F such that K ⊂ F . A space X is called ω-hyperbounded if the closure of any σ-compact subspace of X is compact. We prove that if a space X is (strongly) dominated by an ω-hyperbounded space, then it is (ω-hyperbounded) ω-bounded. We prove that, for a given cardinal κ, if a space X is (strongly) dominated by a κ-hemicompact space, then it is (κ-hemicompact) κ-compact. We analyze relation between domination and hyperspaces of nonempty compact subsets.