Calibers of canonical hyperconnected spaces and their Vietoris hyperspaces

For infinite cardinals $λ\leq κ$, we calculate the calibers of topological spaces $X$ of cardinality $κ$ whose open sets are the subsets $A$ satisfying $|X \setminus A| < λ$. We also prove that the set of calibers of $X$ coincides with the set of calibers of the Vietoris hyperspace $\mathcal{H}(X)$, where the space $\mathcal{H}(X)$ has as its underlying set a collection of subsets of $X$ that includes all its finite subsets and is contained within the collection of its closed subsets.