If a collection of closed sets of arbitrary cardinality in a metric space has empty intersection, does some...












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In this question I claim that every nested sequence of bounded closed subsets of a metric space has nonempty intersection if and only if the space has the Heine-Borel property. However, there's something that can throw a wrench in the proof: what if it is possible for there to be an uncountable collection of closed subsets with empty intersection such that every countable subcollection has nonempty intersection?



Is this possible in a metric space?






general-topology metric-spaces







share|cite|improve this question









$endgroup$

















    5












    $begingroup$


    In this question I claim that every nested sequence of bounded closed subsets of a metric space has nonempty intersection if and only if the space has the Heine-Borel property. However, there's something that can throw a wrench in the proof: what if it is possible for there to be an uncountable collection of closed subsets with empty intersection such that every countable subcollection has nonempty intersection?



    Is this possible in a metric space?






    general-topology metric-spaces







    share|cite|improve this question









    $endgroup$















      5












      5








      5


      1



      $begingroup$


      In this question I claim that every nested sequence of bounded closed subsets of a metric space has nonempty intersection if and only if the space has the Heine-Borel property. However, there's something that can throw a wrench in the proof: what if it is possible for there to be an uncountable collection of closed subsets with empty intersection such that every countable subcollection has nonempty intersection?



      Is this possible in a metric space?






      general-topology metric-spaces







      share|cite|improve this question









      $endgroup$




      In this question I claim that every nested sequence of bounded closed subsets of a metric space has nonempty intersection if and only if the space has the Heine-Borel property. However, there's something that can throw a wrench in the proof: what if it is possible for there to be an uncountable collection of closed subsets with empty intersection such that every countable subcollection has nonempty intersection?



      Is this possible in a metric space?






      general-topology metric-spaces




      general-topology metric-spaces






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Feb 9 at 21:54









      Matt SamuelMatt Samuel

      38.5k63768




      38.5k63768






















          2 Answers
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          7












          $begingroup$

          Let $X$ be an uncountable set endowed with the discrete metric. Then the family ${Xsetminus{x},|,xin X}$ is an uncountable family of closed subsets with empty intersection. But no countable subfamily has that property.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Of course. Thank you.
            $endgroup$
            – Matt Samuel
            Feb 9 at 22:46










          • $begingroup$
            I'm glad I could help.
            $endgroup$
            – José Carlos Santos
            Feb 9 at 23:01



















          6












          $begingroup$

          The property that "if a family of closed sets has an empty intersection, then there is a countable subfamily with empty intersection", has a name. It's called Lindelöf. In a metric space this is equivalent to having a countable dense subset (separable), and many other such countability properties.



          Hence Santos' example was the standard example of a non-separable metric space.






          share|cite|improve this answer









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            2 Answers
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            2 Answers
            2






            active

            oldest

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            active

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            active

            oldest

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            7












            $begingroup$

            Let $X$ be an uncountable set endowed with the discrete metric. Then the family ${Xsetminus{x},|,xin X}$ is an uncountable family of closed subsets with empty intersection. But no countable subfamily has that property.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              Of course. Thank you.
              $endgroup$
              – Matt Samuel
              Feb 9 at 22:46










            • $begingroup$
              I'm glad I could help.
              $endgroup$
              – José Carlos Santos
              Feb 9 at 23:01
















            7












            $begingroup$

            Let $X$ be an uncountable set endowed with the discrete metric. Then the family ${Xsetminus{x},|,xin X}$ is an uncountable family of closed subsets with empty intersection. But no countable subfamily has that property.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              Of course. Thank you.
              $endgroup$
              – Matt Samuel
              Feb 9 at 22:46










            • $begingroup$
              I'm glad I could help.
              $endgroup$
              – José Carlos Santos
              Feb 9 at 23:01














            7












            7








            7





            $begingroup$

            Let $X$ be an uncountable set endowed with the discrete metric. Then the family ${Xsetminus{x},|,xin X}$ is an uncountable family of closed subsets with empty intersection. But no countable subfamily has that property.






            share|cite|improve this answer









            $endgroup$



            Let $X$ be an uncountable set endowed with the discrete metric. Then the family ${Xsetminus{x},|,xin X}$ is an uncountable family of closed subsets with empty intersection. But no countable subfamily has that property.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Feb 9 at 22:11









            José Carlos SantosJosé Carlos Santos

            162k22128232




            162k22128232












            • $begingroup$
              Of course. Thank you.
              $endgroup$
              – Matt Samuel
              Feb 9 at 22:46










            • $begingroup$
              I'm glad I could help.
              $endgroup$
              – José Carlos Santos
              Feb 9 at 23:01


















            • $begingroup$
              Of course. Thank you.
              $endgroup$
              – Matt Samuel
              Feb 9 at 22:46










            • $begingroup$
              I'm glad I could help.
              $endgroup$
              – José Carlos Santos
              Feb 9 at 23:01
















            $begingroup$
            Of course. Thank you.
            $endgroup$
            – Matt Samuel
            Feb 9 at 22:46




            $begingroup$
            Of course. Thank you.
            $endgroup$
            – Matt Samuel
            Feb 9 at 22:46












            $begingroup$
            I'm glad I could help.
            $endgroup$
            – José Carlos Santos
            Feb 9 at 23:01




            $begingroup$
            I'm glad I could help.
            $endgroup$
            – José Carlos Santos
            Feb 9 at 23:01











            6












            $begingroup$

            The property that "if a family of closed sets has an empty intersection, then there is a countable subfamily with empty intersection", has a name. It's called Lindelöf. In a metric space this is equivalent to having a countable dense subset (separable), and many other such countability properties.



            Hence Santos' example was the standard example of a non-separable metric space.






            share|cite|improve this answer









            $endgroup$


















              6












              $begingroup$

              The property that "if a family of closed sets has an empty intersection, then there is a countable subfamily with empty intersection", has a name. It's called Lindelöf. In a metric space this is equivalent to having a countable dense subset (separable), and many other such countability properties.



              Hence Santos' example was the standard example of a non-separable metric space.






              share|cite|improve this answer









              $endgroup$
















                6












                6








                6





                $begingroup$

                The property that "if a family of closed sets has an empty intersection, then there is a countable subfamily with empty intersection", has a name. It's called Lindelöf. In a metric space this is equivalent to having a countable dense subset (separable), and many other such countability properties.



                Hence Santos' example was the standard example of a non-separable metric space.






                share|cite|improve this answer









                $endgroup$



                The property that "if a family of closed sets has an empty intersection, then there is a countable subfamily with empty intersection", has a name. It's called Lindelöf. In a metric space this is equivalent to having a countable dense subset (separable), and many other such countability properties.



                Hence Santos' example was the standard example of a non-separable metric space.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Feb 9 at 23:25









                Henno BrandsmaHenno Brandsma

                110k347116




                110k347116






























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