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Non-Borel set in arbitrary metric space


Derived Sets in arbitrary metric space$A subseteq (X,d)$ is compact. Which metric $p$ makes $(A times A,p)$ also compact and $d: (A times A,p) rightarrow [0,infty)$ continuous?Borel sets and measurabilityapproximate a Borel set by a continuousAn example of Lebesgue measurable set but not Borel measurable besides the “subset of Cantor set” example.A Borel subset of a topological spaceseparability of a metric spacetotally disconnected and non Borel set.What do metric spaces look like?How do we get the notion “Borel regular” measures?













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$begingroup$


Most sources give non-Borel set in Euclidean space. I wonder if there is a way to construct such sets in arbitrary metric space. In particular, is there a non-borel set in $C[0,1]$ all continuous functions on $[0,1]$ where metrics is supremum.










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$endgroup$
















    1












    $begingroup$


    Most sources give non-Borel set in Euclidean space. I wonder if there is a way to construct such sets in arbitrary metric space. In particular, is there a non-borel set in $C[0,1]$ all continuous functions on $[0,1]$ where metrics is supremum.










    share|cite|improve this question









    $endgroup$














      1












      1








      1





      $begingroup$


      Most sources give non-Borel set in Euclidean space. I wonder if there is a way to construct such sets in arbitrary metric space. In particular, is there a non-borel set in $C[0,1]$ all continuous functions on $[0,1]$ where metrics is supremum.










      share|cite|improve this question









      $endgroup$




      Most sources give non-Borel set in Euclidean space. I wonder if there is a way to construct such sets in arbitrary metric space. In particular, is there a non-borel set in $C[0,1]$ all continuous functions on $[0,1]$ where metrics is supremum.







      real-analysis general-topology functional-analysis measure-theory






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      share|cite|improve this question




      share|cite|improve this question










      asked 5 hours ago









      Daniel LiDaniel Li

      752414




      752414




















          2 Answers
          2






          active

          oldest

          votes


















          5












          $begingroup$

          Yes, there is indeed examples of non-Borel sets in $C[0,1]$ of all continuous functions from $[0,1]$ to $mathbbR$ equipped with the uniform norm. Namely, the subset of all continuous nowhere differentiable functions is not a Borel set.



          This result can be found in:
          Mauldin, R. Daniel. The set of continuous nowhere differentiable functions. Pacific J. Math. 83 (1979), no. 1, 199--205.



          In regards to the question on whether it is possible to construct non-Borel sets in arbitrary metric spaces, then the answer is no. Consider the metric space $(x,y,d)$ equipped with the discrete metric $d:x,ytimes x,y to 0,1$ given by
          $$
          d(x,y)=1, quad d(x,x)=d(y,y)=0.
          $$

          The Borel sigma algebra on this metric space is given by
          $$
          x,y,x,y,emptyset = mathcalP(x,y)
          $$

          where $mathcalP(x,y)$ is the powerset of $x,y$, so all subsets are Borel measurable sets.






          share|cite|improve this answer











          $endgroup$












          • $begingroup$
            +1.... With the discrete metric on any set, all subsets are open, and a fortiori, are Borel. Another example would be any countable metric space $X,$ as any $Ysubset X$ is equal to $ cup y:yin Y,$ which is a countable union of closed sets
            $endgroup$
            – DanielWainfleet
            35 mins ago



















          2












          $begingroup$

          Martin gave a specific example in $C[0,1]$ and showed that the general example is negative. Let me argue that a broad class of spaces has a positive answer:



          In any second-countable topological space, there are only continuum-many Borel sets. Since space with at least continuum many points has more than continuum many subsets, this means that every second-countable space with continuum many points has non-Borel subsets.






          share|cite|improve this answer









          $endgroup$












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            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            5












            $begingroup$

            Yes, there is indeed examples of non-Borel sets in $C[0,1]$ of all continuous functions from $[0,1]$ to $mathbbR$ equipped with the uniform norm. Namely, the subset of all continuous nowhere differentiable functions is not a Borel set.



            This result can be found in:
            Mauldin, R. Daniel. The set of continuous nowhere differentiable functions. Pacific J. Math. 83 (1979), no. 1, 199--205.



            In regards to the question on whether it is possible to construct non-Borel sets in arbitrary metric spaces, then the answer is no. Consider the metric space $(x,y,d)$ equipped with the discrete metric $d:x,ytimes x,y to 0,1$ given by
            $$
            d(x,y)=1, quad d(x,x)=d(y,y)=0.
            $$

            The Borel sigma algebra on this metric space is given by
            $$
            x,y,x,y,emptyset = mathcalP(x,y)
            $$

            where $mathcalP(x,y)$ is the powerset of $x,y$, so all subsets are Borel measurable sets.






            share|cite|improve this answer











            $endgroup$












            • $begingroup$
              +1.... With the discrete metric on any set, all subsets are open, and a fortiori, are Borel. Another example would be any countable metric space $X,$ as any $Ysubset X$ is equal to $ cup y:yin Y,$ which is a countable union of closed sets
              $endgroup$
              – DanielWainfleet
              35 mins ago
















            5












            $begingroup$

            Yes, there is indeed examples of non-Borel sets in $C[0,1]$ of all continuous functions from $[0,1]$ to $mathbbR$ equipped with the uniform norm. Namely, the subset of all continuous nowhere differentiable functions is not a Borel set.



            This result can be found in:
            Mauldin, R. Daniel. The set of continuous nowhere differentiable functions. Pacific J. Math. 83 (1979), no. 1, 199--205.



            In regards to the question on whether it is possible to construct non-Borel sets in arbitrary metric spaces, then the answer is no. Consider the metric space $(x,y,d)$ equipped with the discrete metric $d:x,ytimes x,y to 0,1$ given by
            $$
            d(x,y)=1, quad d(x,x)=d(y,y)=0.
            $$

            The Borel sigma algebra on this metric space is given by
            $$
            x,y,x,y,emptyset = mathcalP(x,y)
            $$

            where $mathcalP(x,y)$ is the powerset of $x,y$, so all subsets are Borel measurable sets.






            share|cite|improve this answer











            $endgroup$












            • $begingroup$
              +1.... With the discrete metric on any set, all subsets are open, and a fortiori, are Borel. Another example would be any countable metric space $X,$ as any $Ysubset X$ is equal to $ cup y:yin Y,$ which is a countable union of closed sets
              $endgroup$
              – DanielWainfleet
              35 mins ago














            5












            5








            5





            $begingroup$

            Yes, there is indeed examples of non-Borel sets in $C[0,1]$ of all continuous functions from $[0,1]$ to $mathbbR$ equipped with the uniform norm. Namely, the subset of all continuous nowhere differentiable functions is not a Borel set.



            This result can be found in:
            Mauldin, R. Daniel. The set of continuous nowhere differentiable functions. Pacific J. Math. 83 (1979), no. 1, 199--205.



            In regards to the question on whether it is possible to construct non-Borel sets in arbitrary metric spaces, then the answer is no. Consider the metric space $(x,y,d)$ equipped with the discrete metric $d:x,ytimes x,y to 0,1$ given by
            $$
            d(x,y)=1, quad d(x,x)=d(y,y)=0.
            $$

            The Borel sigma algebra on this metric space is given by
            $$
            x,y,x,y,emptyset = mathcalP(x,y)
            $$

            where $mathcalP(x,y)$ is the powerset of $x,y$, so all subsets are Borel measurable sets.






            share|cite|improve this answer











            $endgroup$



            Yes, there is indeed examples of non-Borel sets in $C[0,1]$ of all continuous functions from $[0,1]$ to $mathbbR$ equipped with the uniform norm. Namely, the subset of all continuous nowhere differentiable functions is not a Borel set.



            This result can be found in:
            Mauldin, R. Daniel. The set of continuous nowhere differentiable functions. Pacific J. Math. 83 (1979), no. 1, 199--205.



            In regards to the question on whether it is possible to construct non-Borel sets in arbitrary metric spaces, then the answer is no. Consider the metric space $(x,y,d)$ equipped with the discrete metric $d:x,ytimes x,y to 0,1$ given by
            $$
            d(x,y)=1, quad d(x,x)=d(y,y)=0.
            $$

            The Borel sigma algebra on this metric space is given by
            $$
            x,y,x,y,emptyset = mathcalP(x,y)
            $$

            where $mathcalP(x,y)$ is the powerset of $x,y$, so all subsets are Borel measurable sets.







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited 4 hours ago

























            answered 4 hours ago









            MartinMartin

            1,106917




            1,106917











            • $begingroup$
              +1.... With the discrete metric on any set, all subsets are open, and a fortiori, are Borel. Another example would be any countable metric space $X,$ as any $Ysubset X$ is equal to $ cup y:yin Y,$ which is a countable union of closed sets
              $endgroup$
              – DanielWainfleet
              35 mins ago

















            • $begingroup$
              +1.... With the discrete metric on any set, all subsets are open, and a fortiori, are Borel. Another example would be any countable metric space $X,$ as any $Ysubset X$ is equal to $ cup y:yin Y,$ which is a countable union of closed sets
              $endgroup$
              – DanielWainfleet
              35 mins ago
















            $begingroup$
            +1.... With the discrete metric on any set, all subsets are open, and a fortiori, are Borel. Another example would be any countable metric space $X,$ as any $Ysubset X$ is equal to $ cup y:yin Y,$ which is a countable union of closed sets
            $endgroup$
            – DanielWainfleet
            35 mins ago





            $begingroup$
            +1.... With the discrete metric on any set, all subsets are open, and a fortiori, are Borel. Another example would be any countable metric space $X,$ as any $Ysubset X$ is equal to $ cup y:yin Y,$ which is a countable union of closed sets
            $endgroup$
            – DanielWainfleet
            35 mins ago












            2












            $begingroup$

            Martin gave a specific example in $C[0,1]$ and showed that the general example is negative. Let me argue that a broad class of spaces has a positive answer:



            In any second-countable topological space, there are only continuum-many Borel sets. Since space with at least continuum many points has more than continuum many subsets, this means that every second-countable space with continuum many points has non-Borel subsets.






            share|cite|improve this answer









            $endgroup$

















              2












              $begingroup$

              Martin gave a specific example in $C[0,1]$ and showed that the general example is negative. Let me argue that a broad class of spaces has a positive answer:



              In any second-countable topological space, there are only continuum-many Borel sets. Since space with at least continuum many points has more than continuum many subsets, this means that every second-countable space with continuum many points has non-Borel subsets.






              share|cite|improve this answer









              $endgroup$















                2












                2








                2





                $begingroup$

                Martin gave a specific example in $C[0,1]$ and showed that the general example is negative. Let me argue that a broad class of spaces has a positive answer:



                In any second-countable topological space, there are only continuum-many Borel sets. Since space with at least continuum many points has more than continuum many subsets, this means that every second-countable space with continuum many points has non-Borel subsets.






                share|cite|improve this answer









                $endgroup$



                Martin gave a specific example in $C[0,1]$ and showed that the general example is negative. Let me argue that a broad class of spaces has a positive answer:



                In any second-countable topological space, there are only continuum-many Borel sets. Since space with at least continuum many points has more than continuum many subsets, this means that every second-countable space with continuum many points has non-Borel subsets.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 3 hours ago









                Noah SchweberNoah Schweber

                127k10151290




                127k10151290



























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