Cardinality of dense set related to cardinality of discrete set












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Let $X$ be a Banach space. Let $eta$ be the density character of $X$ (the least possible cardinality of a dense set). Does there exist $Asubset X$, which is closed and such that $text{card} A=eta$ and $A$ does not have limit points? ($A$ is a discrete set). If this is true, how can we relax the assumptions? It it is false as stated, can we say when it is true? Also, what can we say about the relation between the density character and the cardinality of the maximal discrete subset?






general-topology topological-vector-spaces







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    $begingroup$


    Let $X$ be a Banach space. Let $eta$ be the density character of $X$ (the least possible cardinality of a dense set). Does there exist $Asubset X$, which is closed and such that $text{card} A=eta$ and $A$ does not have limit points? ($A$ is a discrete set). If this is true, how can we relax the assumptions? It it is false as stated, can we say when it is true? Also, what can we say about the relation between the density character and the cardinality of the maximal discrete subset?






    general-topology topological-vector-spaces







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      Let $X$ be a Banach space. Let $eta$ be the density character of $X$ (the least possible cardinality of a dense set). Does there exist $Asubset X$, which is closed and such that $text{card} A=eta$ and $A$ does not have limit points? ($A$ is a discrete set). If this is true, how can we relax the assumptions? It it is false as stated, can we say when it is true? Also, what can we say about the relation between the density character and the cardinality of the maximal discrete subset?






      general-topology topological-vector-spaces







      share|cite|improve this question











      $endgroup$




      Let $X$ be a Banach space. Let $eta$ be the density character of $X$ (the least possible cardinality of a dense set). Does there exist $Asubset X$, which is closed and such that $text{card} A=eta$ and $A$ does not have limit points? ($A$ is a discrete set). If this is true, how can we relax the assumptions? It it is false as stated, can we say when it is true? Also, what can we say about the relation between the density character and the cardinality of the maximal discrete subset?






      general-topology topological-vector-spaces




      general-topology topological-vector-spaces






      share|cite|improve this question















      share|cite|improve this question













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      edited Jan 5 at 22:53







      Stoyan Apostolov

















      asked Jan 5 at 22:42









      Stoyan ApostolovStoyan Apostolov

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          $begingroup$

          In this answer (in particular items 5 and 7 are of interest) I show that the minimal cardinality of a discrete subset and the density character indeed are the same for all metric spaces.



          In particular this holds for Banach spaces. (It's well known that any infinite Hausdorff space at least has one countably-infinite discrete subspace)






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            $begingroup$

            In this answer (in particular items 5 and 7 are of interest) I show that the minimal cardinality of a discrete subset and the density character indeed are the same for all metric spaces.



            In particular this holds for Banach spaces. (It's well known that any infinite Hausdorff space at least has one countably-infinite discrete subspace)






            share|cite|improve this answer









            $endgroup$


















              2












              $begingroup$

              In this answer (in particular items 5 and 7 are of interest) I show that the minimal cardinality of a discrete subset and the density character indeed are the same for all metric spaces.



              In particular this holds for Banach spaces. (It's well known that any infinite Hausdorff space at least has one countably-infinite discrete subspace)






              share|cite|improve this answer









              $endgroup$
















                2












                2








                2





                $begingroup$

                In this answer (in particular items 5 and 7 are of interest) I show that the minimal cardinality of a discrete subset and the density character indeed are the same for all metric spaces.



                In particular this holds for Banach spaces. (It's well known that any infinite Hausdorff space at least has one countably-infinite discrete subspace)






                share|cite|improve this answer









                $endgroup$



                In this answer (in particular items 5 and 7 are of interest) I show that the minimal cardinality of a discrete subset and the density character indeed are the same for all metric spaces.



                In particular this holds for Banach spaces. (It's well known that any infinite Hausdorff space at least has one countably-infinite discrete subspace)







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 5 at 22:50









                Henno BrandsmaHenno Brandsma

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