Domination by second countable spaces and Lindelöf Σ-property

Vietoris topology on spaces dominated by second countable ones

“…The one-point compactification A.Ä/ is an Eberlein compact, then C p .A.Ä// is Lindelöf †. Theorem 2.1 of [5] implies that the function space C p .A.Ä// is a Dieudonné complete and dominated by a second countable space. The hyperspace K.C p .A.Ä/// is also Dieudonné complete (see [16]).…”

“…Suppose that X is strongly dominated by M and take Suppose that K.X / is strongly dominated by the space M . The space X is homeomorphic to the closed subspace F 1 .X/ K.X /, then from 3.3(d) of [5] it follows that X is strongly dominated by the space M . Corollary 2.10.…”

“…Proof. If X is strongly dominated by a space M , then by Theorem 3.3(a) of [5] it follows that Y is strongly dominated by a space M .…”

“…In Theorem 2.1 of [5] it was proved that in the class of Dieudonné complete spaces domination by a second countable space and the Lindelöf † property are equivalent. We will show that in the class of Lindelöf †-spaces strong domination by a second countable space and the Lindelöf p property are not equivalent.…”

“…A space X is strongly dominated by a space M if there exists an M -ordered compact cover F such that for any compact K X there is F 2 F such that K F . Strong domination was introduced by Cascales, Orihuela and Tkachuk in [5].…”