If $E$ is a locally compact separable metric space, then $E$ is the union of a sequence of compact












1












$begingroup$


I have difficulty understanding the following passage.




Let $E$ be a locally compact separable metric space. Let $(x_p)_{pgeq 0}$ be a dense sequence in $E$. Define
$$I:={(p,k)in mathbb{N}^2:bar{B}(x_p,2^{-k}) text{ is compact}},$$
where $bar{B}(x,r)$ is the closed ball with center $x$ and radius $r$. Using the fact that $E$ is locally compact and the density of $(x_p)$ it is easy to see that
$$E=bigcup_{(p,k)in I} bar{B}(x_p,2^{-k}).$$




I don't know why the last sentence is true. If I take $xin E$, then I need to show that $x$ is in some $bar{B}(x_p,2^{-k})$ for $(p,k)in I$. Since $E$ is locally compact, I know that there exists $r>0$ such that $bar{B}(x,r)$ is compact, since $(x_p)$ is dense in $E$, there exist $n$ such that $x_nin bar{B}(x,r)$. But this does not get me to where I need to prove.



I appreciate any help!






general-topology







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    1












    $begingroup$


    I have difficulty understanding the following passage.




    Let $E$ be a locally compact separable metric space. Let $(x_p)_{pgeq 0}$ be a dense sequence in $E$. Define
    $$I:={(p,k)in mathbb{N}^2:bar{B}(x_p,2^{-k}) text{ is compact}},$$
    where $bar{B}(x,r)$ is the closed ball with center $x$ and radius $r$. Using the fact that $E$ is locally compact and the density of $(x_p)$ it is easy to see that
    $$E=bigcup_{(p,k)in I} bar{B}(x_p,2^{-k}).$$




    I don't know why the last sentence is true. If I take $xin E$, then I need to show that $x$ is in some $bar{B}(x_p,2^{-k})$ for $(p,k)in I$. Since $E$ is locally compact, I know that there exists $r>0$ such that $bar{B}(x,r)$ is compact, since $(x_p)$ is dense in $E$, there exist $n$ such that $x_nin bar{B}(x,r)$. But this does not get me to where I need to prove.



    I appreciate any help!






    general-topology







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      I have difficulty understanding the following passage.




      Let $E$ be a locally compact separable metric space. Let $(x_p)_{pgeq 0}$ be a dense sequence in $E$. Define
      $$I:={(p,k)in mathbb{N}^2:bar{B}(x_p,2^{-k}) text{ is compact}},$$
      where $bar{B}(x,r)$ is the closed ball with center $x$ and radius $r$. Using the fact that $E$ is locally compact and the density of $(x_p)$ it is easy to see that
      $$E=bigcup_{(p,k)in I} bar{B}(x_p,2^{-k}).$$




      I don't know why the last sentence is true. If I take $xin E$, then I need to show that $x$ is in some $bar{B}(x_p,2^{-k})$ for $(p,k)in I$. Since $E$ is locally compact, I know that there exists $r>0$ such that $bar{B}(x,r)$ is compact, since $(x_p)$ is dense in $E$, there exist $n$ such that $x_nin bar{B}(x,r)$. But this does not get me to where I need to prove.



      I appreciate any help!






      general-topology







      share|cite|improve this question









      $endgroup$




      I have difficulty understanding the following passage.




      Let $E$ be a locally compact separable metric space. Let $(x_p)_{pgeq 0}$ be a dense sequence in $E$. Define
      $$I:={(p,k)in mathbb{N}^2:bar{B}(x_p,2^{-k}) text{ is compact}},$$
      where $bar{B}(x,r)$ is the closed ball with center $x$ and radius $r$. Using the fact that $E$ is locally compact and the density of $(x_p)$ it is easy to see that
      $$E=bigcup_{(p,k)in I} bar{B}(x_p,2^{-k}).$$




      I don't know why the last sentence is true. If I take $xin E$, then I need to show that $x$ is in some $bar{B}(x_p,2^{-k})$ for $(p,k)in I$. Since $E$ is locally compact, I know that there exists $r>0$ such that $bar{B}(x,r)$ is compact, since $(x_p)$ is dense in $E$, there exist $n$ such that $x_nin bar{B}(x,r)$. But this does not get me to where I need to prove.



      I appreciate any help!






      general-topology




      general-topology






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      asked Feb 3 at 3:41









      JiuJiu

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          $begingroup$

          Let $xin E$ be given.



          Choose $varepsilongt0$ so that $overline B(x,varepsilon)$ is compact.



          Choose $kinmathbb N$ so that $2^{-k}ltfracvarepsilon3$.



          Choose $pinmathbb N$ so that $operatorname d(x_p,x)lt2^{-k}$.



          Then $xinoverline B(x_p,2^{-k})subseteqoverline B(x,varepsilon)$, so $overline B(x_p,2^{-k})$ is compact.






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            $begingroup$

            Let $xin E$ be given.



            Choose $varepsilongt0$ so that $overline B(x,varepsilon)$ is compact.



            Choose $kinmathbb N$ so that $2^{-k}ltfracvarepsilon3$.



            Choose $pinmathbb N$ so that $operatorname d(x_p,x)lt2^{-k}$.



            Then $xinoverline B(x_p,2^{-k})subseteqoverline B(x,varepsilon)$, so $overline B(x_p,2^{-k})$ is compact.






            share|cite|improve this answer









            $endgroup$


















              2












              $begingroup$

              Let $xin E$ be given.



              Choose $varepsilongt0$ so that $overline B(x,varepsilon)$ is compact.



              Choose $kinmathbb N$ so that $2^{-k}ltfracvarepsilon3$.



              Choose $pinmathbb N$ so that $operatorname d(x_p,x)lt2^{-k}$.



              Then $xinoverline B(x_p,2^{-k})subseteqoverline B(x,varepsilon)$, so $overline B(x_p,2^{-k})$ is compact.






              share|cite|improve this answer









              $endgroup$
















                2












                2








                2





                $begingroup$

                Let $xin E$ be given.



                Choose $varepsilongt0$ so that $overline B(x,varepsilon)$ is compact.



                Choose $kinmathbb N$ so that $2^{-k}ltfracvarepsilon3$.



                Choose $pinmathbb N$ so that $operatorname d(x_p,x)lt2^{-k}$.



                Then $xinoverline B(x_p,2^{-k})subseteqoverline B(x,varepsilon)$, so $overline B(x_p,2^{-k})$ is compact.






                share|cite|improve this answer









                $endgroup$



                Let $xin E$ be given.



                Choose $varepsilongt0$ so that $overline B(x,varepsilon)$ is compact.



                Choose $kinmathbb N$ so that $2^{-k}ltfracvarepsilon3$.



                Choose $pinmathbb N$ so that $operatorname d(x_p,x)lt2^{-k}$.



                Then $xinoverline B(x_p,2^{-k})subseteqoverline B(x,varepsilon)$, so $overline B(x_p,2^{-k})$ is compact.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Feb 3 at 3:56









                bofbof

                52.6k559121




                52.6k559121






























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