$C$-embedded and $C^*$-embedded












0












$begingroup$


The set $C(X)$ of all continuous, real-value functions on a topological space $X$ will be provided with an algebraic structure and order structure and the set $C^{*}(X)$ of $C(X)$, consisting of all bounded function in $C(X)$.



A subspace $S$ of $X$ is $C$-embedded in $X$ if every function in $C(S)$ can be extended to a function in $C(X)$. A subspace $S$ of $X$ is $C^{*}$-embedded in $X$ if every function in $C^{*}(S)$ can be extended to a function in $C^{*}(X)$.




My questions are:




1: Does every uniformly continuous function on $mathbb{R} - { 0 }$ have continuous extension to $mathbb{R}$?



2:Is $mathbb{N}$, $C $-embedded or $C^{*}$-embedded in $mathbb{R} $?



According to this theorem that $C^{*}$-embedded is $C$-embedded if only if it is completely separated from every zero-set disjoint from it.



3: Can you give me an example that a $C^{*}$-embedded subspace need not be $C$-embedded?






general-topology functions







share|cite|improve this question











$endgroup$

















    0












    $begingroup$


    The set $C(X)$ of all continuous, real-value functions on a topological space $X$ will be provided with an algebraic structure and order structure and the set $C^{*}(X)$ of $C(X)$, consisting of all bounded function in $C(X)$.



    A subspace $S$ of $X$ is $C$-embedded in $X$ if every function in $C(S)$ can be extended to a function in $C(X)$. A subspace $S$ of $X$ is $C^{*}$-embedded in $X$ if every function in $C^{*}(S)$ can be extended to a function in $C^{*}(X)$.




    My questions are:




    1: Does every uniformly continuous function on $mathbb{R} - { 0 }$ have continuous extension to $mathbb{R}$?



    2:Is $mathbb{N}$, $C $-embedded or $C^{*}$-embedded in $mathbb{R} $?



    According to this theorem that $C^{*}$-embedded is $C$-embedded if only if it is completely separated from every zero-set disjoint from it.



    3: Can you give me an example that a $C^{*}$-embedded subspace need not be $C$-embedded?






    general-topology functions







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      The set $C(X)$ of all continuous, real-value functions on a topological space $X$ will be provided with an algebraic structure and order structure and the set $C^{*}(X)$ of $C(X)$, consisting of all bounded function in $C(X)$.



      A subspace $S$ of $X$ is $C$-embedded in $X$ if every function in $C(S)$ can be extended to a function in $C(X)$. A subspace $S$ of $X$ is $C^{*}$-embedded in $X$ if every function in $C^{*}(S)$ can be extended to a function in $C^{*}(X)$.




      My questions are:




      1: Does every uniformly continuous function on $mathbb{R} - { 0 }$ have continuous extension to $mathbb{R}$?



      2:Is $mathbb{N}$, $C $-embedded or $C^{*}$-embedded in $mathbb{R} $?



      According to this theorem that $C^{*}$-embedded is $C$-embedded if only if it is completely separated from every zero-set disjoint from it.



      3: Can you give me an example that a $C^{*}$-embedded subspace need not be $C$-embedded?






      general-topology functions







      share|cite|improve this question











      $endgroup$




      The set $C(X)$ of all continuous, real-value functions on a topological space $X$ will be provided with an algebraic structure and order structure and the set $C^{*}(X)$ of $C(X)$, consisting of all bounded function in $C(X)$.



      A subspace $S$ of $X$ is $C$-embedded in $X$ if every function in $C(S)$ can be extended to a function in $C(X)$. A subspace $S$ of $X$ is $C^{*}$-embedded in $X$ if every function in $C^{*}(S)$ can be extended to a function in $C^{*}(X)$.




      My questions are:




      1: Does every uniformly continuous function on $mathbb{R} - { 0 }$ have continuous extension to $mathbb{R}$?



      2:Is $mathbb{N}$, $C $-embedded or $C^{*}$-embedded in $mathbb{R} $?



      According to this theorem that $C^{*}$-embedded is $C$-embedded if only if it is completely separated from every zero-set disjoint from it.



      3: Can you give me an example that a $C^{*}$-embedded subspace need not be $C$-embedded?






      general-topology functions




      general-topology functions






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 26 '18 at 18:41









      Namaste

      1




      1










      asked Dec 26 '18 at 17:22







      user387219





























          1 Answer
          1






          active

          oldest

          votes


















          0












          $begingroup$

          (1) Uniform continuity implies that $underset{xrightarrow 0}{lim}f(x)$ exists, and can be extended naturally to $mathbb{R}$.



          (2) Given a bump function $rho$ supported on $(-frac{1}{2},frac{1}{2})$ such that $rho(0)=1$ and $rho(x)=0$ if $vert xvert>frac{1}{2}$, you can define a function:



          $g(x):= sum limits_{n=1}^infty f(n)cdot rho(x+n)$ where $fin C^*(mathbb{N})$.



          Then $g$ extends $f:mathbb{N}rightarrow mathbb{N}$, and $vert g(x)vert leq underset{nin mathbb{N}}{max} vert f(n)vert$, and is also smooth.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Your example fails as your $f$ is itself unbounded on the natural numbers.
            $endgroup$
            – Henno Brandsma
            Dec 26 '18 at 19:09












          • $begingroup$
            As to (1) $infty$ should be $0$ in the limit.
            $endgroup$
            – Henno Brandsma
            Dec 26 '18 at 19:12










          • $begingroup$
            You're right, I'll change (1) and change(2).
            $endgroup$
            – Keen-ameteur
            Dec 26 '18 at 19:16








          • 2




            $begingroup$
            The (3) argument is nonsense. Think again. Examples exist but are non obvious
            $endgroup$
            – Henno Brandsma
            Dec 26 '18 at 19:30










          • $begingroup$
            Can you give me more guidance about 2?
            $endgroup$
            – user387219
            Dec 30 '18 at 10:18











          Your Answer





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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          0












          $begingroup$

          (1) Uniform continuity implies that $underset{xrightarrow 0}{lim}f(x)$ exists, and can be extended naturally to $mathbb{R}$.



          (2) Given a bump function $rho$ supported on $(-frac{1}{2},frac{1}{2})$ such that $rho(0)=1$ and $rho(x)=0$ if $vert xvert>frac{1}{2}$, you can define a function:



          $g(x):= sum limits_{n=1}^infty f(n)cdot rho(x+n)$ where $fin C^*(mathbb{N})$.



          Then $g$ extends $f:mathbb{N}rightarrow mathbb{N}$, and $vert g(x)vert leq underset{nin mathbb{N}}{max} vert f(n)vert$, and is also smooth.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Your example fails as your $f$ is itself unbounded on the natural numbers.
            $endgroup$
            – Henno Brandsma
            Dec 26 '18 at 19:09












          • $begingroup$
            As to (1) $infty$ should be $0$ in the limit.
            $endgroup$
            – Henno Brandsma
            Dec 26 '18 at 19:12










          • $begingroup$
            You're right, I'll change (1) and change(2).
            $endgroup$
            – Keen-ameteur
            Dec 26 '18 at 19:16








          • 2




            $begingroup$
            The (3) argument is nonsense. Think again. Examples exist but are non obvious
            $endgroup$
            – Henno Brandsma
            Dec 26 '18 at 19:30










          • $begingroup$
            Can you give me more guidance about 2?
            $endgroup$
            – user387219
            Dec 30 '18 at 10:18
















          0












          $begingroup$

          (1) Uniform continuity implies that $underset{xrightarrow 0}{lim}f(x)$ exists, and can be extended naturally to $mathbb{R}$.



          (2) Given a bump function $rho$ supported on $(-frac{1}{2},frac{1}{2})$ such that $rho(0)=1$ and $rho(x)=0$ if $vert xvert>frac{1}{2}$, you can define a function:



          $g(x):= sum limits_{n=1}^infty f(n)cdot rho(x+n)$ where $fin C^*(mathbb{N})$.



          Then $g$ extends $f:mathbb{N}rightarrow mathbb{N}$, and $vert g(x)vert leq underset{nin mathbb{N}}{max} vert f(n)vert$, and is also smooth.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Your example fails as your $f$ is itself unbounded on the natural numbers.
            $endgroup$
            – Henno Brandsma
            Dec 26 '18 at 19:09












          • $begingroup$
            As to (1) $infty$ should be $0$ in the limit.
            $endgroup$
            – Henno Brandsma
            Dec 26 '18 at 19:12










          • $begingroup$
            You're right, I'll change (1) and change(2).
            $endgroup$
            – Keen-ameteur
            Dec 26 '18 at 19:16








          • 2




            $begingroup$
            The (3) argument is nonsense. Think again. Examples exist but are non obvious
            $endgroup$
            – Henno Brandsma
            Dec 26 '18 at 19:30










          • $begingroup$
            Can you give me more guidance about 2?
            $endgroup$
            – user387219
            Dec 30 '18 at 10:18














          0












          0








          0





          $begingroup$

          (1) Uniform continuity implies that $underset{xrightarrow 0}{lim}f(x)$ exists, and can be extended naturally to $mathbb{R}$.



          (2) Given a bump function $rho$ supported on $(-frac{1}{2},frac{1}{2})$ such that $rho(0)=1$ and $rho(x)=0$ if $vert xvert>frac{1}{2}$, you can define a function:



          $g(x):= sum limits_{n=1}^infty f(n)cdot rho(x+n)$ where $fin C^*(mathbb{N})$.



          Then $g$ extends $f:mathbb{N}rightarrow mathbb{N}$, and $vert g(x)vert leq underset{nin mathbb{N}}{max} vert f(n)vert$, and is also smooth.






          share|cite|improve this answer











          $endgroup$



          (1) Uniform continuity implies that $underset{xrightarrow 0}{lim}f(x)$ exists, and can be extended naturally to $mathbb{R}$.



          (2) Given a bump function $rho$ supported on $(-frac{1}{2},frac{1}{2})$ such that $rho(0)=1$ and $rho(x)=0$ if $vert xvert>frac{1}{2}$, you can define a function:



          $g(x):= sum limits_{n=1}^infty f(n)cdot rho(x+n)$ where $fin C^*(mathbb{N})$.



          Then $g$ extends $f:mathbb{N}rightarrow mathbb{N}$, and $vert g(x)vert leq underset{nin mathbb{N}}{max} vert f(n)vert$, and is also smooth.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 27 '18 at 5:50

























          answered Dec 26 '18 at 18:34









          Keen-ameteurKeen-ameteur

          1,500516




          1,500516












          • $begingroup$
            Your example fails as your $f$ is itself unbounded on the natural numbers.
            $endgroup$
            – Henno Brandsma
            Dec 26 '18 at 19:09












          • $begingroup$
            As to (1) $infty$ should be $0$ in the limit.
            $endgroup$
            – Henno Brandsma
            Dec 26 '18 at 19:12










          • $begingroup$
            You're right, I'll change (1) and change(2).
            $endgroup$
            – Keen-ameteur
            Dec 26 '18 at 19:16








          • 2




            $begingroup$
            The (3) argument is nonsense. Think again. Examples exist but are non obvious
            $endgroup$
            – Henno Brandsma
            Dec 26 '18 at 19:30










          • $begingroup$
            Can you give me more guidance about 2?
            $endgroup$
            – user387219
            Dec 30 '18 at 10:18


















          • $begingroup$
            Your example fails as your $f$ is itself unbounded on the natural numbers.
            $endgroup$
            – Henno Brandsma
            Dec 26 '18 at 19:09












          • $begingroup$
            As to (1) $infty$ should be $0$ in the limit.
            $endgroup$
            – Henno Brandsma
            Dec 26 '18 at 19:12










          • $begingroup$
            You're right, I'll change (1) and change(2).
            $endgroup$
            – Keen-ameteur
            Dec 26 '18 at 19:16








          • 2




            $begingroup$
            The (3) argument is nonsense. Think again. Examples exist but are non obvious
            $endgroup$
            – Henno Brandsma
            Dec 26 '18 at 19:30










          • $begingroup$
            Can you give me more guidance about 2?
            $endgroup$
            – user387219
            Dec 30 '18 at 10:18
















          $begingroup$
          Your example fails as your $f$ is itself unbounded on the natural numbers.
          $endgroup$
          – Henno Brandsma
          Dec 26 '18 at 19:09






          $begingroup$
          Your example fails as your $f$ is itself unbounded on the natural numbers.
          $endgroup$
          – Henno Brandsma
          Dec 26 '18 at 19:09














          $begingroup$
          As to (1) $infty$ should be $0$ in the limit.
          $endgroup$
          – Henno Brandsma
          Dec 26 '18 at 19:12




          $begingroup$
          As to (1) $infty$ should be $0$ in the limit.
          $endgroup$
          – Henno Brandsma
          Dec 26 '18 at 19:12












          $begingroup$
          You're right, I'll change (1) and change(2).
          $endgroup$
          – Keen-ameteur
          Dec 26 '18 at 19:16






          $begingroup$
          You're right, I'll change (1) and change(2).
          $endgroup$
          – Keen-ameteur
          Dec 26 '18 at 19:16






          2




          2




          $begingroup$
          The (3) argument is nonsense. Think again. Examples exist but are non obvious
          $endgroup$
          – Henno Brandsma
          Dec 26 '18 at 19:30




          $begingroup$
          The (3) argument is nonsense. Think again. Examples exist but are non obvious
          $endgroup$
          – Henno Brandsma
          Dec 26 '18 at 19:30












          $begingroup$
          Can you give me more guidance about 2?
          $endgroup$
          – user387219
          Dec 30 '18 at 10:18




          $begingroup$
          Can you give me more guidance about 2?
          $endgroup$
          – user387219
          Dec 30 '18 at 10:18


















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