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Name for functions with certain boundedness property
The 2019 Stack Overflow Developer Survey Results Are InDoes this property of scattered spaces have a name?Is there a name for relations with this property?Are local quasi-geodesics already quasi-geodesics in hyperbolic spaces?Name for a 'perimeter' metric?Uniform boundedness of functions in non-metrizable spacesIs a continuous function locally uniformly continuous?Is there a name for this property (on sequences)Name for boundedness propertyComplete metric but not uniformly locally compact on the real lineIf Λ is a open subset of a metric space and $K⊆Λ_ε:=x:d(x,Λ^c)>ε$ is compact, is there a compact $L⊆Λ_ε$ s.t. $B_δ(x)⊆L$ for all $x∈K$?
$begingroup$
Let $f: X_1 rightarrow X_2 $ be a function between two metric spaces. My question is, if there is a name in the standard literature for the following property of $f$ in $x in X_1$:
$$(1)~~~~~~~~~~~ forall varepsilon > 0 ~~exists delta > 0 :~~operatornameim_f(B_varepsilon(x) ) subseteq B_delta(f(x))$$
i.e. the image of every $varepsilon$-ball is bounded.
By looking for obvious candidates for a name, I only found the term of ,,local boundedness in $x$'' which can in this case be expressed as
$$(2)~~~~~~~~~~~ exists varepsilon > 0 ~~exists delta > 0 :~~operatornameim_vert f vert(B_varepsilon(x) ) subseteq B_delta(0)$$
They are of course not equivalent, since the for example: the real function $f(x)=x^-1$ is locally bounded in every point $x neq 0$ but does not have property $(1)$ in any point.
general-topology functions metric-spaces terminology
asked Mar 30 at 10:48
NemoNemo
854519
$endgroup$
|
show 1 more comment
$begingroup$
Let $f: X_1 rightarrow X_2 $ be a function between two metric spaces. My question is, if there is a name in the standard literature for the following property of $f$ in $x in X_1$:
$$(1)~~~~~~~~~~~ forall varepsilon > 0 ~~exists delta > 0 :~~operatornameim_f(B_varepsilon(x) ) subseteq B_delta(f(x))$$
i.e. the image of every $varepsilon$-ball is bounded.
By looking for obvious candidates for a name, I only found the term of ,,local boundedness in $x$'' which can in this case be expressed as
$$(2)~~~~~~~~~~~ exists varepsilon > 0 ~~exists delta > 0 :~~operatornameim_vert f vert(B_varepsilon(x) ) subseteq B_delta(0)$$
They are of course not equivalent, since the for example: the real function $f(x)=x^-1$ is locally bounded in every point $x neq 0$ but does not have property $(1)$ in any point.
general-topology functions metric-spaces terminology
asked Mar 30 at 10:48
NemoNemo
854519
$endgroup$
$begingroup$
Is the $delta$ mentioned global? That is, is the condition $forallepsilon>0,,existsdelta>0,,forall xin X_1ldots$ the condition you are intending, or does that $delta$ depend on $x$?
$endgroup$
– Robert Thingum
Mar 31 at 0:03
$begingroup$
A consequence of your definition (regardless of whether or not the $delta$ is global) is that if $f:X_1rightarrow X_2$ is such a map then $f(X_1)subseteq B(f(x),delta_x)$ for every $xin X_1$. Not sure if you want this or not.
$endgroup$
– Robert Thingum
Mar 31 at 0:06
$begingroup$
@RobertThingum No, $delta$ is not meant to be global. So the quantification was more intended to be $forall x in X_1 ~forall varepsilon ~exists delta ~...$, meanig $delta = delta(varepsilon,x)$.
$endgroup$
– Nemo
Mar 31 at 12:24
$begingroup$
@RobertThingum I think $f(X_1) subseteq B(f(x) , delta_x)$ is only possible if $X_1$ is bounded. It doesn't work out in the example I gave, since $f(mathbbR setminus 0 ) = mathbbR setminus 0 $.
$endgroup$
– Nemo
Mar 31 at 12:30
$begingroup$
Your functions look like bornologous functions. en.wikipedia.org/wiki/Bornological_space#Bounded_maps
$endgroup$
– Robert Thingum
Mar 31 at 13:29
|
show 1 more comment
$begingroup$
Let $f: X_1 rightarrow X_2 $ be a function between two metric spaces. My question is, if there is a name in the standard literature for the following property of $f$ in $x in X_1$:
$$(1)~~~~~~~~~~~ forall varepsilon > 0 ~~exists delta > 0 :~~operatornameim_f(B_varepsilon(x) ) subseteq B_delta(f(x))$$
i.e. the image of every $varepsilon$-ball is bounded.
By looking for obvious candidates for a name, I only found the term of ,,local boundedness in $x$'' which can in this case be expressed as
$$(2)~~~~~~~~~~~ exists varepsilon > 0 ~~exists delta > 0 :~~operatornameim_vert f vert(B_varepsilon(x) ) subseteq B_delta(0)$$
They are of course not equivalent, since the for example: the real function $f(x)=x^-1$ is locally bounded in every point $x neq 0$ but does not have property $(1)$ in any point.
general-topology functions metric-spaces terminology
asked Mar 30 at 10:48
NemoNemo
854519
$endgroup$
Let $f: X_1 rightarrow X_2 $ be a function between two metric spaces. My question is, if there is a name in the standard literature for the following property of $f$ in $x in X_1$:
$$(1)~~~~~~~~~~~ forall varepsilon > 0 ~~exists delta > 0 :~~operatornameim_f(B_varepsilon(x) ) subseteq B_delta(f(x))$$
i.e. the image of every $varepsilon$-ball is bounded.
By looking for obvious candidates for a name, I only found the term of ,,local boundedness in $x$'' which can in this case be expressed as
$$(2)~~~~~~~~~~~ exists varepsilon > 0 ~~exists delta > 0 :~~operatornameim_vert f vert(B_varepsilon(x) ) subseteq B_delta(0)$$
They are of course not equivalent, since the for example: the real function $f(x)=x^-1$ is locally bounded in every point $x neq 0$ but does not have property $(1)$ in any point.
general-topology functions metric-spaces terminology
general-topology functions metric-spaces terminology
asked Mar 30 at 10:48
NemoNemo
854519
asked Mar 30 at 10:48
NemoNemo
854519
edited Mar 30 at 20:27
Nemo
asked Mar 30 at 10:48
NemoNemo
854519
asked Mar 30 at 10:48
NemoNemo
854519
asked Mar 30 at 10:48
NemoNemo
854519
854519
$begingroup$
Is the $delta$ mentioned global? That is, is the condition $forallepsilon>0,,existsdelta>0,,forall xin X_1ldots$ the condition you are intending, or does that $delta$ depend on $x$?
$endgroup$
– Robert Thingum
Mar 31 at 0:03
$begingroup$
A consequence of your definition (regardless of whether or not the $delta$ is global) is that if $f:X_1rightarrow X_2$ is such a map then $f(X_1)subseteq B(f(x),delta_x)$ for every $xin X_1$. Not sure if you want this or not.
$endgroup$
– Robert Thingum
Mar 31 at 0:06
$begingroup$
@RobertThingum No, $delta$ is not meant to be global. So the quantification was more intended to be $forall x in X_1 ~forall varepsilon ~exists delta ~...$, meanig $delta = delta(varepsilon,x)$.
$endgroup$
– Nemo
Mar 31 at 12:24
$begingroup$
@RobertThingum I think $f(X_1) subseteq B(f(x) , delta_x)$ is only possible if $X_1$ is bounded. It doesn't work out in the example I gave, since $f(mathbbR setminus 0 ) = mathbbR setminus 0 $.
$endgroup$
– Nemo
Mar 31 at 12:30
$begingroup$
Your functions look like bornologous functions. en.wikipedia.org/wiki/Bornological_space#Bounded_maps
$endgroup$
– Robert Thingum
Mar 31 at 13:29
|
show 1 more comment
$begingroup$
Is the $delta$ mentioned global? That is, is the condition $forallepsilon>0,,existsdelta>0,,forall xin X_1ldots$ the condition you are intending, or does that $delta$ depend on $x$?
$endgroup$
– Robert Thingum
Mar 31 at 0:03
$begingroup$
A consequence of your definition (regardless of whether or not the $delta$ is global) is that if $f:X_1rightarrow X_2$ is such a map then $f(X_1)subseteq B(f(x),delta_x)$ for every $xin X_1$. Not sure if you want this or not.
$endgroup$
– Robert Thingum
Mar 31 at 0:06
$begingroup$
@RobertThingum No, $delta$ is not meant to be global. So the quantification was more intended to be $forall x in X_1 ~forall varepsilon ~exists delta ~...$, meanig $delta = delta(varepsilon,x)$.
$endgroup$
– Nemo
Mar 31 at 12:24
$begingroup$
@RobertThingum I think $f(X_1) subseteq B(f(x) , delta_x)$ is only possible if $X_1$ is bounded. It doesn't work out in the example I gave, since $f(mathbbR setminus 0 ) = mathbbR setminus 0 $.
$endgroup$
– Nemo
Mar 31 at 12:30
$begingroup$
Your functions look like bornologous functions. en.wikipedia.org/wiki/Bornological_space#Bounded_maps
$endgroup$
– Robert Thingum
Mar 31 at 13:29
$begingroup$
Is the $delta$ mentioned global? That is, is the condition $forallepsilon>0,,existsdelta>0,,forall xin X_1ldots$ the condition you are intending, or does that $delta$ depend on $x$?
$endgroup$
– Robert Thingum
Mar 31 at 0:03
$begingroup$
Is the $delta$ mentioned global? That is, is the condition $forallepsilon>0,,existsdelta>0,,forall xin X_1ldots$ the condition you are intending, or does that $delta$ depend on $x$?
$endgroup$
– Robert Thingum
Mar 31 at 0:03
$begingroup$
A consequence of your definition (regardless of whether or not the $delta$ is global) is that if $f:X_1rightarrow X_2$ is such a map then $f(X_1)subseteq B(f(x),delta_x)$ for every $xin X_1$. Not sure if you want this or not.
$endgroup$
– Robert Thingum
Mar 31 at 0:06
$begingroup$
A consequence of your definition (regardless of whether or not the $delta$ is global) is that if $f:X_1rightarrow X_2$ is such a map then $f(X_1)subseteq B(f(x),delta_x)$ for every $xin X_1$. Not sure if you want this or not.
$endgroup$
– Robert Thingum
Mar 31 at 0:06
$begingroup$
@RobertThingum No, $delta$ is not meant to be global. So the quantification was more intended to be $forall x in X_1 ~forall varepsilon ~exists delta ~...$, meanig $delta = delta(varepsilon,x)$.
$endgroup$
– Nemo
Mar 31 at 12:24
$begingroup$
@RobertThingum No, $delta$ is not meant to be global. So the quantification was more intended to be $forall x in X_1 ~forall varepsilon ~exists delta ~...$, meanig $delta = delta(varepsilon,x)$.
$endgroup$
– Nemo
Mar 31 at 12:24
$begingroup$
@RobertThingum I think $f(X_1) subseteq B(f(x) , delta_x)$ is only possible if $X_1$ is bounded. It doesn't work out in the example I gave, since $f(mathbbR setminus 0 ) = mathbbR setminus 0 $.
$endgroup$
– Nemo
Mar 31 at 12:30
$begingroup$
@RobertThingum I think $f(X_1) subseteq B(f(x) , delta_x)$ is only possible if $X_1$ is bounded. It doesn't work out in the example I gave, since $f(mathbbR setminus 0 ) = mathbbR setminus 0 $.
$endgroup$
– Nemo
Mar 31 at 12:30
$begingroup$
Your functions look like bornologous functions. en.wikipedia.org/wiki/Bornological_space#Bounded_maps
$endgroup$
– Robert Thingum
Mar 31 at 13:29
$begingroup$
Your functions look like bornologous functions. en.wikipedia.org/wiki/Bornological_space#Bounded_maps
$endgroup$
– Robert Thingum
Mar 31 at 13:29
|
show 1 more comment
1 Answer
1
active
oldest
votes
$begingroup$
The functions satisfying $(1)$ are the functions sending metrically bounded sets of $X_1$ to metrically bounded sets of $X_2$.
See the following
Bornological Space on wiki
If you are interesting in reading about some applications of these kinds of functions within an active area of research you should look into coarse geometry.
Coarse Spaces on wiki
edited Mar 31 at 13:57
Alex Vong
1,340819
answered Mar 31 at 13:24
Robert ThingumRobert Thingum
9811317
$endgroup$
$begingroup$
Is it right to say $f$ preserves metrically bounded sets?
$endgroup$
– Alex Vong
Mar 31 at 13:44
$begingroup$
I would say so. You could also say that $f$ simply preserves "boundedness".
$endgroup$
– Robert Thingum
Mar 31 at 13:45
add a comment |
Your Answer
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The functions satisfying $(1)$ are the functions sending metrically bounded sets of $X_1$ to metrically bounded sets of $X_2$.
See the following
Bornological Space on wiki
If you are interesting in reading about some applications of these kinds of functions within an active area of research you should look into coarse geometry.
Coarse Spaces on wiki
edited Mar 31 at 13:57
Alex Vong
1,340819
answered Mar 31 at 13:24
Robert ThingumRobert Thingum
9811317
$endgroup$
$begingroup$
Is it right to say $f$ preserves metrically bounded sets?
$endgroup$
– Alex Vong
Mar 31 at 13:44
$begingroup$
I would say so. You could also say that $f$ simply preserves "boundedness".
$endgroup$
– Robert Thingum
Mar 31 at 13:45
add a comment |
$begingroup$
The functions satisfying $(1)$ are the functions sending metrically bounded sets of $X_1$ to metrically bounded sets of $X_2$.
See the following
Bornological Space on wiki
If you are interesting in reading about some applications of these kinds of functions within an active area of research you should look into coarse geometry.
Coarse Spaces on wiki
edited Mar 31 at 13:57
Alex Vong
1,340819
answered Mar 31 at 13:24
Robert ThingumRobert Thingum
9811317
$endgroup$
$begingroup$
Is it right to say $f$ preserves metrically bounded sets?
$endgroup$
– Alex Vong
Mar 31 at 13:44
$begingroup$
I would say so. You could also say that $f$ simply preserves "boundedness".
$endgroup$
– Robert Thingum
Mar 31 at 13:45
add a comment |
$begingroup$
The functions satisfying $(1)$ are the functions sending metrically bounded sets of $X_1$ to metrically bounded sets of $X_2$.
See the following
Bornological Space on wiki
If you are interesting in reading about some applications of these kinds of functions within an active area of research you should look into coarse geometry.
Coarse Spaces on wiki
edited Mar 31 at 13:57
Alex Vong
1,340819
answered Mar 31 at 13:24
Robert ThingumRobert Thingum
9811317
$endgroup$
The functions satisfying $(1)$ are the functions sending metrically bounded sets of $X_1$ to metrically bounded sets of $X_2$.
See the following
Bornological Space on wiki
If you are interesting in reading about some applications of these kinds of functions within an active area of research you should look into coarse geometry.
Coarse Spaces on wiki
edited Mar 31 at 13:57
Alex Vong
1,340819
answered Mar 31 at 13:24
Robert ThingumRobert Thingum
9811317
edited Mar 31 at 13:57
Alex Vong
1,340819
edited Mar 31 at 13:57
Alex Vong
1,340819
edited Mar 31 at 13:57
Alex Vong
1,340819
1,340819
answered Mar 31 at 13:24
Robert ThingumRobert Thingum
9811317
answered Mar 31 at 13:24
Robert ThingumRobert Thingum
9811317
answered Mar 31 at 13:24
Robert ThingumRobert Thingum
9811317
9811317
$begingroup$
Is it right to say $f$ preserves metrically bounded sets?
$endgroup$
– Alex Vong
Mar 31 at 13:44
$begingroup$
I would say so. You could also say that $f$ simply preserves "boundedness".
$endgroup$
– Robert Thingum
Mar 31 at 13:45
add a comment |
$begingroup$
Is it right to say $f$ preserves metrically bounded sets?
$endgroup$
– Alex Vong
Mar 31 at 13:44
$begingroup$
I would say so. You could also say that $f$ simply preserves "boundedness".
$endgroup$
– Robert Thingum
Mar 31 at 13:45
$begingroup$
Is it right to say $f$ preserves metrically bounded sets?
$endgroup$
– Alex Vong
Mar 31 at 13:44
$begingroup$
Is it right to say $f$ preserves metrically bounded sets?
$endgroup$
– Alex Vong
Mar 31 at 13:44
$begingroup$
I would say so. You could also say that $f$ simply preserves "boundedness".
$endgroup$
– Robert Thingum
Mar 31 at 13:45
$begingroup$
I would say so. You could also say that $f$ simply preserves "boundedness".
$endgroup$
– Robert Thingum
Mar 31 at 13:45
add a comment |
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$begingroup$
Is the $delta$ mentioned global? That is, is the condition $forallepsilon>0,,existsdelta>0,,forall xin X_1ldots$ the condition you are intending, or does that $delta$ depend on $x$?
$endgroup$
– Robert Thingum
Mar 31 at 0:03
$begingroup$
A consequence of your definition (regardless of whether or not the $delta$ is global) is that if $f:X_1rightarrow X_2$ is such a map then $f(X_1)subseteq B(f(x),delta_x)$ for every $xin X_1$. Not sure if you want this or not.
$endgroup$
– Robert Thingum
Mar 31 at 0:06
$begingroup$
@RobertThingum No, $delta$ is not meant to be global. So the quantification was more intended to be $forall x in X_1 ~forall varepsilon ~exists delta ~...$, meanig $delta = delta(varepsilon,x)$.
$endgroup$
– Nemo
Mar 31 at 12:24
$begingroup$
@RobertThingum I think $f(X_1) subseteq B(f(x) , delta_x)$ is only possible if $X_1$ is bounded. It doesn't work out in the example I gave, since $f(mathbbR setminus 0 ) = mathbbR setminus 0 $.
$endgroup$
– Nemo
Mar 31 at 12:30
$begingroup$
Your functions look like bornologous functions. en.wikipedia.org/wiki/Bornological_space#Bounded_maps
$endgroup$
– Robert Thingum
Mar 31 at 13:29