On the Fuzzy Number Space with the Level Convergence Topology

Abstract

We characterize compact sets of 𝔼1 endowed with the level convergence topology τℓ. We also describe the completion (𝔼1̂,𝒰̂) of 𝔼1 with respect to its natural uniformity, that is, the pointwise uniformity 𝒰, and show other topological properties of 𝔼1̂, as separability. We apply these results to give an Arzela-Ascoli theorem for the space of (𝔼1,τℓ)-valued continuous functions on a locally compact topological space equipped with the compact-open topology.