Compactness and local compactness in hyperspaces

Dynamic Equations on Time Scales

“…The statement that it is locally compact turned out to contain an error -a correction to the problematic proposition can be found in [5] -however it is now known that CL(R) is not locally compact. A result of Ivanova, Keesling and Velichko says that if the Vietoris topology on CL(X) is normal, then X is compact: see [10], [11], and [16].…”

On complete metrizability of the Hausdorff metric topology

“…It follows that a i ∈ A i [a i−1 ] for all 1 ≤ i, thus, p = (a i ) i≥0 ∈ P and s p (k) = a m (k) for all k < 2n m − 1. This implies that d(a m , S) ≤ d(a m , s p ) ≤ In conclusion, by (12), (14) we have that H d (A m , S) < 1 nm , thus, S ∈ V m , which implies that S ∈ m<ω V m , and so α wins.…”

“…Since complete metrizability of a metrizable space is equivalent to itš Cech-completeness (i.e. being G δ in a compactification [17]), the recent characterization of local compactness of (CL(X), τ H d ) by Costantini, Levi, Pelant in [12,Corollary 15], as well as of the intermediary property of cofinal completeness of (CL(X), τ H d ) by Beer, Di Maio in [6, Theorem 3.9.] must be mentioned here, as they both imply complete metrizability of (CL(X), τ H d ).…”

Compactness and local compactness of the proximal hyperspace

“…In [2] it has been proved that local compactness of the Vietoris hyperspace is in general a strictly stronger condition than local compactness of the Hausdor hyperspace. As a consequence of Corollary 3.5, local compactness of the proximal hyperspace has the same behaviour.…”