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Let $Y$ and $Z$ be completions of $X$, then there exists an isometry from $Y$ onto $Z$.
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Proof Attempt: Let $X$ be dense on metric spaces $Y$ and $Z$.
Let $overlineX=Y$ and $overlineX=Z$. Suppose that $a_n$ is a sequence in $A$ such that $a_nrightarrow y,z$ such that $yin Y$ and $zin Z$. Let $MinBbbN$ be such that $forall igeq M$, we have $d_Y(y,a_i),d_Z(z,a_i)ltepsilon/2$. Then, we extend the identity mapping on $X$, by defining $i(y)=z$. Hence, we have
beginalign
vert d_Y(y,a_i)-d_Z(i(y),i(a_i))vert=vert d_Y(y,a_i)-d_Z(z,a_i)vertltepsilon
endalign
This means $forall yin Y$, $forall zin Z$, and for all sequences $a_nrightarrow y,z$ in $X$, we have $d_Y(y,a_i)=d_Z(z,a_i)$ - an isometry.
proof-verification metric-spaces isometry
asked Aug 1 at 9:07
TheLast Cipher
538414
 |Â
show 3 more comments
up vote
0
down vote
favorite
Proof Attempt: Let $X$ be dense on metric spaces $Y$ and $Z$.
Let $overlineX=Y$ and $overlineX=Z$. Suppose that $a_n$ is a sequence in $A$ such that $a_nrightarrow y,z$ such that $yin Y$ and $zin Z$. Let $MinBbbN$ be such that $forall igeq M$, we have $d_Y(y,a_i),d_Z(z,a_i)ltepsilon/2$. Then, we extend the identity mapping on $X$, by defining $i(y)=z$. Hence, we have
beginalign
vert d_Y(y,a_i)-d_Z(i(y),i(a_i))vert=vert d_Y(y,a_i)-d_Z(z,a_i)vertltepsilon
endalign
This means $forall yin Y$, $forall zin Z$, and for all sequences $a_nrightarrow y,z$ in $X$, we have $d_Y(y,a_i)=d_Z(z,a_i)$ - an isometry.
proof-verification metric-spaces isometry
asked Aug 1 at 9:07
TheLast Cipher
538414
How do you define $overline X = Y$?
– Kenny Lau
Aug 1 at 9:07
I just edited. I meant closure of the metric space.
– TheLast Cipher
Aug 1 at 9:09
How do you define closure?
– Kenny Lau
Aug 1 at 9:09
set of all limit points of the space.
– TheLast Cipher
Aug 1 at 9:11
I don't think that is quite well-defined.
– Kenny Lau
Aug 1 at 9:11
 |Â
show 3 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Proof Attempt: Let $X$ be dense on metric spaces $Y$ and $Z$.
Let $overlineX=Y$ and $overlineX=Z$. Suppose that $a_n$ is a sequence in $A$ such that $a_nrightarrow y,z$ such that $yin Y$ and $zin Z$. Let $MinBbbN$ be such that $forall igeq M$, we have $d_Y(y,a_i),d_Z(z,a_i)ltepsilon/2$. Then, we extend the identity mapping on $X$, by defining $i(y)=z$. Hence, we have
beginalign
vert d_Y(y,a_i)-d_Z(i(y),i(a_i))vert=vert d_Y(y,a_i)-d_Z(z,a_i)vertltepsilon
endalign
This means $forall yin Y$, $forall zin Z$, and for all sequences $a_nrightarrow y,z$ in $X$, we have $d_Y(y,a_i)=d_Z(z,a_i)$ - an isometry.
proof-verification metric-spaces isometry
asked Aug 1 at 9:07
TheLast Cipher
538414
Proof Attempt: Let $X$ be dense on metric spaces $Y$ and $Z$.
Let $overlineX=Y$ and $overlineX=Z$. Suppose that $a_n$ is a sequence in $A$ such that $a_nrightarrow y,z$ such that $yin Y$ and $zin Z$. Let $MinBbbN$ be such that $forall igeq M$, we have $d_Y(y,a_i),d_Z(z,a_i)ltepsilon/2$. Then, we extend the identity mapping on $X$, by defining $i(y)=z$. Hence, we have
beginalign
vert d_Y(y,a_i)-d_Z(i(y),i(a_i))vert=vert d_Y(y,a_i)-d_Z(z,a_i)vertltepsilon
endalign
This means $forall yin Y$, $forall zin Z$, and for all sequences $a_nrightarrow y,z$ in $X$, we have $d_Y(y,a_i)=d_Z(z,a_i)$ - an isometry.
proof-verification metric-spaces isometry
asked Aug 1 at 9:07
TheLast Cipher
538414
edited Aug 1 at 9:13
asked Aug 1 at 9:07
TheLast Cipher
538414
asked Aug 1 at 9:07
TheLast Cipher
538414
asked Aug 1 at 9:07
TheLast Cipher
538414
538414
How do you define $overline X = Y$?
– Kenny Lau
Aug 1 at 9:07
I just edited. I meant closure of the metric space.
– TheLast Cipher
Aug 1 at 9:09
How do you define closure?
– Kenny Lau
Aug 1 at 9:09
set of all limit points of the space.
– TheLast Cipher
Aug 1 at 9:11
I don't think that is quite well-defined.
– Kenny Lau
Aug 1 at 9:11
 |Â
show 3 more comments
How do you define $overline X = Y$?
– Kenny Lau
Aug 1 at 9:07
I just edited. I meant closure of the metric space.
– TheLast Cipher
Aug 1 at 9:09
How do you define closure?
– Kenny Lau
Aug 1 at 9:09
set of all limit points of the space.
– TheLast Cipher
Aug 1 at 9:11
I don't think that is quite well-defined.
– Kenny Lau
Aug 1 at 9:11
How do you define $overline X = Y$?
– Kenny Lau
Aug 1 at 9:07
How do you define $overline X = Y$?
– Kenny Lau
Aug 1 at 9:07
I just edited. I meant closure of the metric space.
– TheLast Cipher
Aug 1 at 9:09
I just edited. I meant closure of the metric space.
– TheLast Cipher
Aug 1 at 9:09
How do you define closure?
– Kenny Lau
Aug 1 at 9:09
How do you define closure?
– Kenny Lau
Aug 1 at 9:09
set of all limit points of the space.
– TheLast Cipher
Aug 1 at 9:11
set of all limit points of the space.
– TheLast Cipher
Aug 1 at 9:11
I don't think that is quite well-defined.
– Kenny Lau
Aug 1 at 9:11
I don't think that is quite well-defined.
– Kenny Lau
Aug 1 at 9:11
 |Â
show 3 more comments
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How do you define $overline X = Y$?
– Kenny Lau
Aug 1 at 9:07
I just edited. I meant closure of the metric space.
– TheLast Cipher
Aug 1 at 9:09
How do you define closure?
– Kenny Lau
Aug 1 at 9:09
set of all limit points of the space.
– TheLast Cipher
Aug 1 at 9:11
I don't think that is quite well-defined.
– Kenny Lau
Aug 1 at 9:11