Hausdorff partial metric
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Using a metric on a metric space $(X,d)$ we can define Hausdorff metric $h$ on the set of all compact subsets of $(X,d)$,say $K(X)$. I am able to show that $h$ is indeed a metric. Suppose instead of metric $d$ on $X$ we had a partial metric $p$ on $X$. If we analogously define Hausdorff partial metric, say $h_p$, similar to the definition of $h$, will $h_p$ be a partial metric non $K(X)$?
Edit
Let $Xne emptyset$. A partial metric on $X$ is a function $p:Xtimes X to mathbb{R}^+$ satisfying
a)$p(x,y)=p(y,x)$ (symmetry)
b)$0le p(x,x) =p(x,y)= p(y,y)$ then $x=y$ (equality)
c)$p(x,x)le p(x,y)$ (small self distances)
d)$p(x,z)le p(x,y)+p(y,z)-p(y,y)$ (triangle inequality)
for all $x,y,zin X.$
Why partial metric is useful?
Some research articles state that, partial metric is useful in modeling partially defined information as comes in computer science.
Definition
$$h_p(A,B)=sup_{ain A}inf_{bin B} p(a,b)$$
$K(X)$: set of all compact subsets in the partial metric space $(X,p)$
Question
Is $h_p$ a partial metric on $K(X)$?
general-topology metric-spaces compactness fractals hausdorff-distance
asked Nov 17 at 7:54
supremum
1,2131719
add a comment |
up vote
1
down vote
favorite
Using a metric on a metric space $(X,d)$ we can define Hausdorff metric $h$ on the set of all compact subsets of $(X,d)$,say $K(X)$. I am able to show that $h$ is indeed a metric. Suppose instead of metric $d$ on $X$ we had a partial metric $p$ on $X$. If we analogously define Hausdorff partial metric, say $h_p$, similar to the definition of $h$, will $h_p$ be a partial metric non $K(X)$?
Edit
Let $Xne emptyset$. A partial metric on $X$ is a function $p:Xtimes X to mathbb{R}^+$ satisfying
a)$p(x,y)=p(y,x)$ (symmetry)
b)$0le p(x,x) =p(x,y)= p(y,y)$ then $x=y$ (equality)
c)$p(x,x)le p(x,y)$ (small self distances)
d)$p(x,z)le p(x,y)+p(y,z)-p(y,y)$ (triangle inequality)
for all $x,y,zin X.$
Why partial metric is useful?
Some research articles state that, partial metric is useful in modeling partially defined information as comes in computer science.
Definition
$$h_p(A,B)=sup_{ain A}inf_{bin B} p(a,b)$$
$K(X)$: set of all compact subsets in the partial metric space $(X,p)$
Question
Is $h_p$ a partial metric on $K(X)$?
general-topology metric-spaces compactness fractals hausdorff-distance
asked Nov 17 at 7:54
supremum
1,2131719
define partial metric, and why is that notion useful at all?
– Henno Brandsma
Nov 17 at 9:18
@HennoBrandsma On partial metric spaces: dcs.warwick.ac.uk/pmetric/index.html
– supremum
Nov 17 at 10:28
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Using a metric on a metric space $(X,d)$ we can define Hausdorff metric $h$ on the set of all compact subsets of $(X,d)$,say $K(X)$. I am able to show that $h$ is indeed a metric. Suppose instead of metric $d$ on $X$ we had a partial metric $p$ on $X$. If we analogously define Hausdorff partial metric, say $h_p$, similar to the definition of $h$, will $h_p$ be a partial metric non $K(X)$?
Edit
Let $Xne emptyset$. A partial metric on $X$ is a function $p:Xtimes X to mathbb{R}^+$ satisfying
a)$p(x,y)=p(y,x)$ (symmetry)
b)$0le p(x,x) =p(x,y)= p(y,y)$ then $x=y$ (equality)
c)$p(x,x)le p(x,y)$ (small self distances)
d)$p(x,z)le p(x,y)+p(y,z)-p(y,y)$ (triangle inequality)
for all $x,y,zin X.$
Why partial metric is useful?
Some research articles state that, partial metric is useful in modeling partially defined information as comes in computer science.
Definition
$$h_p(A,B)=sup_{ain A}inf_{bin B} p(a,b)$$
$K(X)$: set of all compact subsets in the partial metric space $(X,p)$
Question
Is $h_p$ a partial metric on $K(X)$?
general-topology metric-spaces compactness fractals hausdorff-distance
asked Nov 17 at 7:54
supremum
1,2131719
Using a metric on a metric space $(X,d)$ we can define Hausdorff metric $h$ on the set of all compact subsets of $(X,d)$,say $K(X)$. I am able to show that $h$ is indeed a metric. Suppose instead of metric $d$ on $X$ we had a partial metric $p$ on $X$. If we analogously define Hausdorff partial metric, say $h_p$, similar to the definition of $h$, will $h_p$ be a partial metric non $K(X)$?
Edit
Let $Xne emptyset$. A partial metric on $X$ is a function $p:Xtimes X to mathbb{R}^+$ satisfying
a)$p(x,y)=p(y,x)$ (symmetry)
b)$0le p(x,x) =p(x,y)= p(y,y)$ then $x=y$ (equality)
c)$p(x,x)le p(x,y)$ (small self distances)
d)$p(x,z)le p(x,y)+p(y,z)-p(y,y)$ (triangle inequality)
for all $x,y,zin X.$
Why partial metric is useful?
Some research articles state that, partial metric is useful in modeling partially defined information as comes in computer science.
Definition
$$h_p(A,B)=sup_{ain A}inf_{bin B} p(a,b)$$
$K(X)$: set of all compact subsets in the partial metric space $(X,p)$
Question
Is $h_p$ a partial metric on $K(X)$?
general-topology metric-spaces compactness fractals hausdorff-distance
general-topology metric-spaces compactness fractals hausdorff-distance
asked Nov 17 at 7:54
supremum
1,2131719
asked Nov 17 at 7:54
supremum
1,2131719
edited Nov 17 at 10:09
asked Nov 17 at 7:54
supremum
1,2131719
asked Nov 17 at 7:54
supremum
1,2131719
asked Nov 17 at 7:54
supremum
1,2131719
1,2131719
define partial metric, and why is that notion useful at all?
– Henno Brandsma
Nov 17 at 9:18
@HennoBrandsma On partial metric spaces: dcs.warwick.ac.uk/pmetric/index.html
– supremum
Nov 17 at 10:28
add a comment |
define partial metric, and why is that notion useful at all?
– Henno Brandsma
Nov 17 at 9:18
@HennoBrandsma On partial metric spaces: dcs.warwick.ac.uk/pmetric/index.html
– supremum
Nov 17 at 10:28
define partial metric, and why is that notion useful at all?
– Henno Brandsma
Nov 17 at 9:18
define partial metric, and why is that notion useful at all?
– Henno Brandsma
Nov 17 at 9:18
@HennoBrandsma On partial metric spaces: dcs.warwick.ac.uk/pmetric/index.html
– supremum
Nov 17 at 10:28
@HennoBrandsma On partial metric spaces: dcs.warwick.ac.uk/pmetric/index.html
– supremum
Nov 17 at 10:28
add a comment |
1 Answer
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Is $h_p$ a partial metric on $K(X)$?
Not necessarily. Let $X$ be the first of the examples of partial metric space from the site linked by you, that is $X=Bbb R^+$ with $p(x,y)=max{x}$. Let $A={1,2,3}$ and $B={1,3}$ be elements of $K(X)$. Then $$h_p(A,A)= h_p(A,B)= h_p(B,B),$$ but $Ane B$, which violates the equality of $h_p$.
answered Dec 5 at 13:08
Alex Ravsky
37.9k32079
add a comment |
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1 Answer
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Is $h_p$ a partial metric on $K(X)$?
Not necessarily. Let $X$ be the first of the examples of partial metric space from the site linked by you, that is $X=Bbb R^+$ with $p(x,y)=max{x}$. Let $A={1,2,3}$ and $B={1,3}$ be elements of $K(X)$. Then $$h_p(A,A)= h_p(A,B)= h_p(B,B),$$ but $Ane B$, which violates the equality of $h_p$.
answered Dec 5 at 13:08
Alex Ravsky
37.9k32079
add a comment |
up vote
0
down vote
Is $h_p$ a partial metric on $K(X)$?
Not necessarily. Let $X$ be the first of the examples of partial metric space from the site linked by you, that is $X=Bbb R^+$ with $p(x,y)=max{x}$. Let $A={1,2,3}$ and $B={1,3}$ be elements of $K(X)$. Then $$h_p(A,A)= h_p(A,B)= h_p(B,B),$$ but $Ane B$, which violates the equality of $h_p$.
answered Dec 5 at 13:08
Alex Ravsky
37.9k32079
add a comment |
up vote
0
down vote
up vote
0
down vote
Is $h_p$ a partial metric on $K(X)$?
Not necessarily. Let $X$ be the first of the examples of partial metric space from the site linked by you, that is $X=Bbb R^+$ with $p(x,y)=max{x}$. Let $A={1,2,3}$ and $B={1,3}$ be elements of $K(X)$. Then $$h_p(A,A)= h_p(A,B)= h_p(B,B),$$ but $Ane B$, which violates the equality of $h_p$.
answered Dec 5 at 13:08
Alex Ravsky
37.9k32079
Is $h_p$ a partial metric on $K(X)$?
Not necessarily. Let $X$ be the first of the examples of partial metric space from the site linked by you, that is $X=Bbb R^+$ with $p(x,y)=max{x}$. Let $A={1,2,3}$ and $B={1,3}$ be elements of $K(X)$. Then $$h_p(A,A)= h_p(A,B)= h_p(B,B),$$ but $Ane B$, which violates the equality of $h_p$.
answered Dec 5 at 13:08
Alex Ravsky
37.9k32079
answered Dec 5 at 13:08
Alex Ravsky
37.9k32079
answered Dec 5 at 13:08
Alex Ravsky
37.9k32079
answered Dec 5 at 13:08
Alex Ravsky
37.9k32079
37.9k32079
add a comment |
add a comment |
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define partial metric, and why is that notion useful at all?
– Henno Brandsma
Nov 17 at 9:18
@HennoBrandsma On partial metric spaces: dcs.warwick.ac.uk/pmetric/index.html
– supremum
Nov 17 at 10:28