Uniformizing (proximal) Δ-topologies

Beer and Tamaki investigated necessary and sufficient conditions for the uniformizability of (proximal) Δ-topologies. proofs involved construction of special Urysohn functions. In this paper we attack the same problem using as a useful tool a uniform topology with reference to a Hausdorff uniformity patterned after the one related to the Attouch–Wets topology. We also study ΔU-topologies, proximal ΔU-topologies which are natural generalizations of the U-topology discovered by Costantini and Vitolo.