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What is the compactness uniqueness argument
0
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I have come across the term "compactness uniqueness argument" in many inverse problems papers and books but the proof are usually omitted. Could anyone give such an example and relevant references?
compactness inverse-problems
edited Jan 9 at 21:46
David G. Stork
10.9k31432
asked Jan 9 at 21:43
KKKKKK
1658
$endgroup$
add a comment |
0
$begingroup$
I have come across the term "compactness uniqueness argument" in many inverse problems papers and books but the proof are usually omitted. Could anyone give such an example and relevant references?
compactness inverse-problems
edited Jan 9 at 21:46
David G. Stork
10.9k31432
asked Jan 9 at 21:43
KKKKKK
1658
$endgroup$
1
$begingroup$
What kind of statement (what kind of uniqueness?) is this sort of argument supposed to prove?
$endgroup$
– Mindlack
Jan 9 at 21:59
1
$begingroup$
I don't think this can be answered unless you show or link or cite at least one (complete) example. The only things that come to my mind are (1). If $X, Y$ are compact Hausdorff spaces and $f:Xto Y$ is a continuous bijection then $f$ is a homeomorphism... (2). If $T,U$ are compact Hausdorff topologies on $X$ and $Usubset T$ , then $U=T$. Because $id_X$ is a continuous bijection from $(X, T)$ to $(X,U)$ so by (1), $ id_X $ is a homeomorphism from $(X,T)$ to $ (X,U)$.
$endgroup$
– DanielWainfleet
Jan 10 at 0:54
$begingroup$
Could you give an example of a reference to where the argument is used?
$endgroup$
– Tommi Brander
Jan 10 at 10:47
add a comment |
0
0
0
$begingroup$
I have come across the term "compactness uniqueness argument" in many inverse problems papers and books but the proof are usually omitted. Could anyone give such an example and relevant references?
compactness inverse-problems
edited Jan 9 at 21:46
David G. Stork
10.9k31432
asked Jan 9 at 21:43
KKKKKK
1658
$endgroup$
I have come across the term "compactness uniqueness argument" in many inverse problems papers and books but the proof are usually omitted. Could anyone give such an example and relevant references?
compactness inverse-problems
compactness inverse-problems
edited Jan 9 at 21:46
David G. Stork
10.9k31432
asked Jan 9 at 21:43
KKKKKK
1658
edited Jan 9 at 21:46
David G. Stork
10.9k31432
asked Jan 9 at 21:43
KKKKKK
1658
edited Jan 9 at 21:46
David G. Stork
10.9k31432
edited Jan 9 at 21:46
David G. Stork
10.9k31432
edited Jan 9 at 21:46
David G. Stork
10.9k31432
10.9k31432
asked Jan 9 at 21:43
KKKKKK
1658
asked Jan 9 at 21:43
KKKKKK
1658
asked Jan 9 at 21:43
KKKKKK
1658
1658
1
$begingroup$
What kind of statement (what kind of uniqueness?) is this sort of argument supposed to prove?
$endgroup$
– Mindlack
Jan 9 at 21:59
1
$begingroup$
I don't think this can be answered unless you show or link or cite at least one (complete) example. The only things that come to my mind are (1). If $X, Y$ are compact Hausdorff spaces and $f:Xto Y$ is a continuous bijection then $f$ is a homeomorphism... (2). If $T,U$ are compact Hausdorff topologies on $X$ and $Usubset T$ , then $U=T$. Because $id_X$ is a continuous bijection from $(X, T)$ to $(X,U)$ so by (1), $ id_X $ is a homeomorphism from $(X,T)$ to $ (X,U)$.
$endgroup$
– DanielWainfleet
Jan 10 at 0:54
$begingroup$
Could you give an example of a reference to where the argument is used?
$endgroup$
– Tommi Brander
Jan 10 at 10:47
add a comment |
1
$begingroup$
What kind of statement (what kind of uniqueness?) is this sort of argument supposed to prove?
$endgroup$
– Mindlack
Jan 9 at 21:59
1
$begingroup$
I don't think this can be answered unless you show or link or cite at least one (complete) example. The only things that come to my mind are (1). If $X, Y$ are compact Hausdorff spaces and $f:Xto Y$ is a continuous bijection then $f$ is a homeomorphism... (2). If $T,U$ are compact Hausdorff topologies on $X$ and $Usubset T$ , then $U=T$. Because $id_X$ is a continuous bijection from $(X, T)$ to $(X,U)$ so by (1), $ id_X $ is a homeomorphism from $(X,T)$ to $ (X,U)$.
$endgroup$
– DanielWainfleet
Jan 10 at 0:54
$begingroup$
Could you give an example of a reference to where the argument is used?
$endgroup$
– Tommi Brander
Jan 10 at 10:47
1
1
$begingroup$
What kind of statement (what kind of uniqueness?) is this sort of argument supposed to prove?
$endgroup$
– Mindlack
Jan 9 at 21:59
$begingroup$
What kind of statement (what kind of uniqueness?) is this sort of argument supposed to prove?
$endgroup$
– Mindlack
Jan 9 at 21:59
1
1
$begingroup$
I don't think this can be answered unless you show or link or cite at least one (complete) example. The only things that come to my mind are (1). If $X, Y$ are compact Hausdorff spaces and $f:Xto Y$ is a continuous bijection then $f$ is a homeomorphism... (2). If $T,U$ are compact Hausdorff topologies on $X$ and $Usubset T$ , then $U=T$. Because $id_X$ is a continuous bijection from $(X, T)$ to $(X,U)$ so by (1), $ id_X $ is a homeomorphism from $(X,T)$ to $ (X,U)$.
$endgroup$
– DanielWainfleet
Jan 10 at 0:54
$begingroup$
I don't think this can be answered unless you show or link or cite at least one (complete) example. The only things that come to my mind are (1). If $X, Y$ are compact Hausdorff spaces and $f:Xto Y$ is a continuous bijection then $f$ is a homeomorphism... (2). If $T,U$ are compact Hausdorff topologies on $X$ and $Usubset T$ , then $U=T$. Because $id_X$ is a continuous bijection from $(X, T)$ to $(X,U)$ so by (1), $ id_X $ is a homeomorphism from $(X,T)$ to $ (X,U)$.
$endgroup$
– DanielWainfleet
Jan 10 at 0:54
$begingroup$
Could you give an example of a reference to where the argument is used?
$endgroup$
– Tommi Brander
Jan 10 at 10:47
$begingroup$
Could you give an example of a reference to where the argument is used?
$endgroup$
– Tommi Brander
Jan 10 at 10:47
add a comment |
0
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1
$begingroup$
What kind of statement (what kind of uniqueness?) is this sort of argument supposed to prove?
$endgroup$
– Mindlack
Jan 9 at 21:59
1
$begingroup$
I don't think this can be answered unless you show or link or cite at least one (complete) example. The only things that come to my mind are (1). If $X, Y$ are compact Hausdorff spaces and $f:Xto Y$ is a continuous bijection then $f$ is a homeomorphism... (2). If $T,U$ are compact Hausdorff topologies on $X$ and $Usubset T$ , then $U=T$. Because $id_X$ is a continuous bijection from $(X, T)$ to $(X,U)$ so by (1), $ id_X $ is a homeomorphism from $(X,T)$ to $ (X,U)$.
$endgroup$
– DanielWainfleet
Jan 10 at 0:54
$begingroup$
Could you give an example of a reference to where the argument is used?
$endgroup$
– Tommi Brander
Jan 10 at 10:47