Extension of a metric defined on a closed subset
$begingroup$
If $X$ is any metrizable space, $A$ is a closed subset of $X$.
Let $d$ be a compatible metric on $A$
then $d$ can be extended to a compatible metric on $X$.
real-analysis general-topology metric-spaces
edited Jul 7 '13 at 11:30
Hagen von Eitzen
284k23274508
asked Jul 7 '13 at 11:15
akanshaakansha
800514
$endgroup$
add a comment |
$begingroup$
If $X$ is any metrizable space, $A$ is a closed subset of $X$.
Let $d$ be a compatible metric on $A$
then $d$ can be extended to a compatible metric on $X$.
real-analysis general-topology metric-spaces
edited Jul 7 '13 at 11:30
Hagen von Eitzen
284k23274508
asked Jul 7 '13 at 11:15
akanshaakansha
800514
$endgroup$
2
$begingroup$
and the question is?
$endgroup$
– Emanuele Paolini
Jul 7 '13 at 11:16
add a comment |
$begingroup$
If $X$ is any metrizable space, $A$ is a closed subset of $X$.
Let $d$ be a compatible metric on $A$
then $d$ can be extended to a compatible metric on $X$.
real-analysis general-topology metric-spaces
edited Jul 7 '13 at 11:30
Hagen von Eitzen
284k23274508
asked Jul 7 '13 at 11:15
akanshaakansha
800514
$endgroup$
If $X$ is any metrizable space, $A$ is a closed subset of $X$.
Let $d$ be a compatible metric on $A$
then $d$ can be extended to a compatible metric on $X$.
real-analysis general-topology metric-spaces
real-analysis general-topology metric-spaces
edited Jul 7 '13 at 11:30
Hagen von Eitzen
284k23274508
asked Jul 7 '13 at 11:15
akanshaakansha
800514
edited Jul 7 '13 at 11:30
Hagen von Eitzen
284k23274508
asked Jul 7 '13 at 11:15
akanshaakansha
800514
edited Jul 7 '13 at 11:30
Hagen von Eitzen
284k23274508
edited Jul 7 '13 at 11:30
Hagen von Eitzen
284k23274508
edited Jul 7 '13 at 11:30
Hagen von Eitzen
284k23274508
284k23274508
asked Jul 7 '13 at 11:15
akanshaakansha
800514
asked Jul 7 '13 at 11:15
akanshaakansha
800514
asked Jul 7 '13 at 11:15
akanshaakansha
800514
800514
2
$begingroup$
and the question is?
$endgroup$
– Emanuele Paolini
Jul 7 '13 at 11:16
add a comment |
2
$begingroup$
and the question is?
$endgroup$
– Emanuele Paolini
Jul 7 '13 at 11:16
2
2
$begingroup$
and the question is?
$endgroup$
– Emanuele Paolini
Jul 7 '13 at 11:16
$begingroup$
and the question is?
$endgroup$
– Emanuele Paolini
Jul 7 '13 at 11:16
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I cite (with a correction) the beginning of my paper “On Extension of (Pseudo-)Metrics from Subgroup of Topological Group onto the Group”:
“The problem of extensions of functions from subobjects to objects in
various categories was considered by many authors. The classic
Tietze-Urysohn theorem on extensions of functions from a closed subspace of
a topological space and its generalizations belong to the known results.
Hausdorff [3] showed that every metric from a closed subspace of
a metrizable space can be extended onto the space. Isbell [4, Lemma 1.4] showed that every bounded uniformly continuous pseudometric on a subspace of a uniform space can be extended to a bounded uniformly continuous pseudometric on the whole space. The linear operators extending metrics from a closed subspace of a metrizable space onto the space were considered in, e.g.,
[2,6]”.
References
[2] Bessaga C., Functional analytic aspects of geometry. Linear extending of metrics and related problems, in: Progress of Functional
Analysis, Proc. Peniscola Meeting 1990 on the 60th birthday of Professor M. Valdivia, North-Holland, Amsterdam (1992) 247-257.
[3] Hausdorff F. Erweiterung einer Homömorpie, - Fund. Math., 16 (1930) 353-360.
[4] Isbell J.R. On finite-dimensional uniform spaces, - Pacific
J. of Math., 9 (1959) 107-121.
[6] Zarichnyi M., Regular Linear Operators Extending Metrics: a Short Proof,
Bull. Pol. Ac.:Math., 44 (1996) 267-269.
answered Jul 7 '13 at 13:09
Alex RavskyAlex Ravsky
43.7k32584
$endgroup$
1
$begingroup$
Re :"Isbell [4] showed that..." This doesn't look right.What if the subspace is empty? Not all uniform space are metrizable.
$endgroup$
– DanielWainfleet
Sep 10 '15 at 20:44
$begingroup$
@DanielWainfleet Thanks. I corrected it and added a link to the paper.
$endgroup$
– Alex Ravsky
Jan 4 at 14:07
add a comment |
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1 Answer
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active
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$begingroup$
I cite (with a correction) the beginning of my paper “On Extension of (Pseudo-)Metrics from Subgroup of Topological Group onto the Group”:
“The problem of extensions of functions from subobjects to objects in
various categories was considered by many authors. The classic
Tietze-Urysohn theorem on extensions of functions from a closed subspace of
a topological space and its generalizations belong to the known results.
Hausdorff [3] showed that every metric from a closed subspace of
a metrizable space can be extended onto the space. Isbell [4, Lemma 1.4] showed that every bounded uniformly continuous pseudometric on a subspace of a uniform space can be extended to a bounded uniformly continuous pseudometric on the whole space. The linear operators extending metrics from a closed subspace of a metrizable space onto the space were considered in, e.g.,
[2,6]”.
References
[2] Bessaga C., Functional analytic aspects of geometry. Linear extending of metrics and related problems, in: Progress of Functional
Analysis, Proc. Peniscola Meeting 1990 on the 60th birthday of Professor M. Valdivia, North-Holland, Amsterdam (1992) 247-257.
[3] Hausdorff F. Erweiterung einer Homömorpie, - Fund. Math., 16 (1930) 353-360.
[4] Isbell J.R. On finite-dimensional uniform spaces, - Pacific
J. of Math., 9 (1959) 107-121.
[6] Zarichnyi M., Regular Linear Operators Extending Metrics: a Short Proof,
Bull. Pol. Ac.:Math., 44 (1996) 267-269.
answered Jul 7 '13 at 13:09
Alex RavskyAlex Ravsky
43.7k32584
$endgroup$
1
$begingroup$
Re :"Isbell [4] showed that..." This doesn't look right.What if the subspace is empty? Not all uniform space are metrizable.
$endgroup$
– DanielWainfleet
Sep 10 '15 at 20:44
$begingroup$
@DanielWainfleet Thanks. I corrected it and added a link to the paper.
$endgroup$
– Alex Ravsky
Jan 4 at 14:07
add a comment |
$begingroup$
I cite (with a correction) the beginning of my paper “On Extension of (Pseudo-)Metrics from Subgroup of Topological Group onto the Group”:
“The problem of extensions of functions from subobjects to objects in
various categories was considered by many authors. The classic
Tietze-Urysohn theorem on extensions of functions from a closed subspace of
a topological space and its generalizations belong to the known results.
Hausdorff [3] showed that every metric from a closed subspace of
a metrizable space can be extended onto the space. Isbell [4, Lemma 1.4] showed that every bounded uniformly continuous pseudometric on a subspace of a uniform space can be extended to a bounded uniformly continuous pseudometric on the whole space. The linear operators extending metrics from a closed subspace of a metrizable space onto the space were considered in, e.g.,
[2,6]”.
References
[2] Bessaga C., Functional analytic aspects of geometry. Linear extending of metrics and related problems, in: Progress of Functional
Analysis, Proc. Peniscola Meeting 1990 on the 60th birthday of Professor M. Valdivia, North-Holland, Amsterdam (1992) 247-257.
[3] Hausdorff F. Erweiterung einer Homömorpie, - Fund. Math., 16 (1930) 353-360.
[4] Isbell J.R. On finite-dimensional uniform spaces, - Pacific
J. of Math., 9 (1959) 107-121.
[6] Zarichnyi M., Regular Linear Operators Extending Metrics: a Short Proof,
Bull. Pol. Ac.:Math., 44 (1996) 267-269.
answered Jul 7 '13 at 13:09
Alex RavskyAlex Ravsky
43.7k32584
$endgroup$
1
$begingroup$
Re :"Isbell [4] showed that..." This doesn't look right.What if the subspace is empty? Not all uniform space are metrizable.
$endgroup$
– DanielWainfleet
Sep 10 '15 at 20:44
$begingroup$
@DanielWainfleet Thanks. I corrected it and added a link to the paper.
$endgroup$
– Alex Ravsky
Jan 4 at 14:07
add a comment |
$begingroup$
I cite (with a correction) the beginning of my paper “On Extension of (Pseudo-)Metrics from Subgroup of Topological Group onto the Group”:
“The problem of extensions of functions from subobjects to objects in
various categories was considered by many authors. The classic
Tietze-Urysohn theorem on extensions of functions from a closed subspace of
a topological space and its generalizations belong to the known results.
Hausdorff [3] showed that every metric from a closed subspace of
a metrizable space can be extended onto the space. Isbell [4, Lemma 1.4] showed that every bounded uniformly continuous pseudometric on a subspace of a uniform space can be extended to a bounded uniformly continuous pseudometric on the whole space. The linear operators extending metrics from a closed subspace of a metrizable space onto the space were considered in, e.g.,
[2,6]”.
References
[2] Bessaga C., Functional analytic aspects of geometry. Linear extending of metrics and related problems, in: Progress of Functional
Analysis, Proc. Peniscola Meeting 1990 on the 60th birthday of Professor M. Valdivia, North-Holland, Amsterdam (1992) 247-257.
[3] Hausdorff F. Erweiterung einer Homömorpie, - Fund. Math., 16 (1930) 353-360.
[4] Isbell J.R. On finite-dimensional uniform spaces, - Pacific
J. of Math., 9 (1959) 107-121.
[6] Zarichnyi M., Regular Linear Operators Extending Metrics: a Short Proof,
Bull. Pol. Ac.:Math., 44 (1996) 267-269.
answered Jul 7 '13 at 13:09
Alex RavskyAlex Ravsky
43.7k32584
$endgroup$
I cite (with a correction) the beginning of my paper “On Extension of (Pseudo-)Metrics from Subgroup of Topological Group onto the Group”:
“The problem of extensions of functions from subobjects to objects in
various categories was considered by many authors. The classic
Tietze-Urysohn theorem on extensions of functions from a closed subspace of
a topological space and its generalizations belong to the known results.
Hausdorff [3] showed that every metric from a closed subspace of
a metrizable space can be extended onto the space. Isbell [4, Lemma 1.4] showed that every bounded uniformly continuous pseudometric on a subspace of a uniform space can be extended to a bounded uniformly continuous pseudometric on the whole space. The linear operators extending metrics from a closed subspace of a metrizable space onto the space were considered in, e.g.,
[2,6]”.
References
[2] Bessaga C., Functional analytic aspects of geometry. Linear extending of metrics and related problems, in: Progress of Functional
Analysis, Proc. Peniscola Meeting 1990 on the 60th birthday of Professor M. Valdivia, North-Holland, Amsterdam (1992) 247-257.
[3] Hausdorff F. Erweiterung einer Homömorpie, - Fund. Math., 16 (1930) 353-360.
[4] Isbell J.R. On finite-dimensional uniform spaces, - Pacific
J. of Math., 9 (1959) 107-121.
[6] Zarichnyi M., Regular Linear Operators Extending Metrics: a Short Proof,
Bull. Pol. Ac.:Math., 44 (1996) 267-269.
answered Jul 7 '13 at 13:09
Alex RavskyAlex Ravsky
43.7k32584
edited Jan 4 at 14:05
answered Jul 7 '13 at 13:09
Alex RavskyAlex Ravsky
43.7k32584
answered Jul 7 '13 at 13:09
Alex RavskyAlex Ravsky
43.7k32584
answered Jul 7 '13 at 13:09
Alex RavskyAlex Ravsky
43.7k32584
43.7k32584
1
$begingroup$
Re :"Isbell [4] showed that..." This doesn't look right.What if the subspace is empty? Not all uniform space are metrizable.
$endgroup$
– DanielWainfleet
Sep 10 '15 at 20:44
$begingroup$
@DanielWainfleet Thanks. I corrected it and added a link to the paper.
$endgroup$
– Alex Ravsky
Jan 4 at 14:07
add a comment |
1
$begingroup$
Re :"Isbell [4] showed that..." This doesn't look right.What if the subspace is empty? Not all uniform space are metrizable.
$endgroup$
– DanielWainfleet
Sep 10 '15 at 20:44
$begingroup$
@DanielWainfleet Thanks. I corrected it and added a link to the paper.
$endgroup$
– Alex Ravsky
Jan 4 at 14:07
1
1
$begingroup$
Re :"Isbell [4] showed that..." This doesn't look right.What if the subspace is empty? Not all uniform space are metrizable.
$endgroup$
– DanielWainfleet
Sep 10 '15 at 20:44
$begingroup$
Re :"Isbell [4] showed that..." This doesn't look right.What if the subspace is empty? Not all uniform space are metrizable.
$endgroup$
– DanielWainfleet
Sep 10 '15 at 20:44
$begingroup$
@DanielWainfleet Thanks. I corrected it and added a link to the paper.
$endgroup$
– Alex Ravsky
Jan 4 at 14:07
$begingroup$
@DanielWainfleet Thanks. I corrected it and added a link to the paper.
$endgroup$
– Alex Ravsky
Jan 4 at 14:07
add a comment |
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$begingroup$
and the question is?
$endgroup$
– Emanuele Paolini
Jul 7 '13 at 11:16