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Recently I come across the description of "a space topologized by convergence in probability". And I can search that it is the topology generated by a metric defined by [tex]E\left[ \frac{|X|}{1+|X|}\right][/tex].

I can understand how a topology related to a metric. However, whenever we come across the word "topologize by some convergence", does that implicitly mean that we should work out the corresponding metric defined for such convergence and then generate the topology by using this metric?

Lastly, can anyone explain why there is no topology associated with a.s. convergence since I cannot find any reference regarding this? Does this mean there is no well-defined metric for a.s. convergence?

Thanks.

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# Topologize by convergence

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