From a topological point of view, it is usually best to work with the weakest topology having some desired property. In this way, no extraneous conditions on convergence are posed. Thus, while it seems interesting to study stronger topologies, I don’t think they could replace the weaker ones.

Topologies are not always meant to make sense of convergence. For instance, the KL divergence can be used to determine the exponential rate of convergence of the likelihood ratio. This fact is used in bayesian stat. to show that priors assigning positive mass to KL neighborhoods are weakly consistent a posteriori. Replacing the KL topology by any other would only weaken this result.

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