Sunday, 17 April 2016

real analysis - bijective measurable map existence



Does there exist bijective measurable maps between R and Rn?



If so, could you give me an example of that?



Thank you.


Answer




Polish space is a topological space that is isomorphic to a complete separable metric space, for example Rn for any nN. For the proof of the following fact, see e.g. here.




Any uncountable Polish space is Borel isomorphic (there exists a bimeasurable bijection) to the space of real numbers R with standard topology.



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