$C$-embedded and $C^*$-embedded
$begingroup$
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
edited Dec 26 '18 at 18:41
Namaste
1
$endgroup$
add a comment |
$begingroup$
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
edited Dec 26 '18 at 18:41
Namaste
1
$endgroup$
add a comment |
$begingroup$
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
edited Dec 26 '18 at 18:41
Namaste
1
$endgroup$
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
edited Dec 26 '18 at 18:41
Namaste
1
edited Dec 26 '18 at 18:41
Namaste
1
edited Dec 26 '18 at 18:41
Namaste
1
edited Dec 26 '18 at 18:41
Namaste
1
edited Dec 26 '18 at 18:41
Namaste
1
1
asked Dec 26 '18 at 17:22
user387219
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
(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.
answered Dec 26 '18 at 18:34
Keen-ameteurKeen-ameteur
1,500516
$endgroup$
$begingroup$
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
|
show 1 more comment
Your Answer
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
(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.
answered Dec 26 '18 at 18:34
Keen-ameteurKeen-ameteur
1,500516
$endgroup$
$begingroup$
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
|
show 1 more comment
$begingroup$
(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.
answered Dec 26 '18 at 18:34
Keen-ameteurKeen-ameteur
1,500516
$endgroup$
$begingroup$
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
|
show 1 more comment
$begingroup$
(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.
answered Dec 26 '18 at 18:34
Keen-ameteurKeen-ameteur
1,500516
$endgroup$
(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.
answered Dec 26 '18 at 18:34
Keen-ameteurKeen-ameteur
1,500516
edited Dec 27 '18 at 5:50
answered Dec 26 '18 at 18:34
Keen-ameteurKeen-ameteur
1,500516
answered Dec 26 '18 at 18:34
Keen-ameteurKeen-ameteur
1,500516
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
|
show 1 more comment
$begingroup$
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
|
show 1 more comment
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