Weighted topologies on Ran(M) interpolate Hausdorff and final topologies, equip the latter with a complete uniformity, and are conically stratified when M is Riemannian.
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Old and new structures on Ran spaces: Length structures, completeness, and conicality
Weighted topologies on Ran(M) interpolate Hausdorff and final topologies, equip the latter with a complete uniformity, and are conically stratified when M is Riemannian.