Existence of selections and disconnectedness properties for the hyperspace of an ultrametric space

Orderability and continuous selections for Wijsman and Vietoris hyperspaces

“…This yields to the following result, which completes the main one of [3] and shows that the converse of Proposition 8.9 is also valid provided that d is an ultrametric.…”

“…Dealing with the problem of finding conditions equivalent to the existence of continuous selections for the Wijsman topology, Bertacchi and Costantini introduce the following notion (see Definition 2 in [3]), which turns out to play a fundamental role in such a research: Definition 8.1. Let (X, d) be a metric space and x ∈ X.…”

“…The fact that condition ( ) is sufficient for the Wijsman hyperspace to admit a continuous selection, stated by Bertacchi and Costantini in the setting of ultrametric spaces (see (a) ⇒ (b) of Proposition 8.2), follows now as an easy consequence and without requiring d to be an ultrametric. This also solves the problem which is implicitely raised in the last two lines of [3].…”

“…Acknowledgements. The authors wish to thank C. Costantini for remarking that Corollary 8.14 actually solves the problem left open in [3] and suggesting Proposition 8.15, as well as the comments before it.…”

Vietoris continuous selections and disconnectedness-like properties

“…In this way, we get a τ Vclopen neighbourhood g −1 (U ) of F in F (X). Let M ⊂ g −1 (U ) be a chain which is maximal with respect to the usual set-theoretical inclusion and F ∈ M. Then, there exists M = max M because g −1 (U ) is τ V -closed (see [2], [4], [6]). Indeed, it suffices to show that M = M ∈ g −1 (U ).…”