On Upper Dini’s Systems and U.S.C. Functions With Convex Limit Sets - Holý, Dušan; Matejička, Ladislav (Sarajevo Journal of Mathematics, 2024)
We give an answer to the question in [HN] as to which upper Dini's systems of functions induces a Hausdorff metric topology on $U_0(X)$. We show that if $X$ is a locally connected metric space then the Hausdorff metric topology on $U_0(X)$ induces as an upper Dini's system of functions the set of all bounded upper semicontinuous functions vanishing at infinity with convex limit sets. 2000 Mathematics Subject Classification. Primary: 54C35; secondary: 54C60