Condition for equality of “distance to the boundary” and “distance to the set”
Let $(X,d)$ be a metric space, $A subseteq X$ and $x notin A$. What are the condition which make true the statement
$d(x,A) = d(x,partial A)$
If A is compact (not empty) I can think of a way to find a point $y in A$ such that $d(x,y)=d(x,A)$. And if $X$ is a normed space that should be enough to prove $y in partial A$. Is there something beside this?
metric-spaces
asked 7 hours ago
Markus Steiner
735
add a comment |
Let $(X,d)$ be a metric space, $A subseteq X$ and $x notin A$. What are the condition which make true the statement
$d(x,A) = d(x,partial A)$
If A is compact (not empty) I can think of a way to find a point $y in A$ such that $d(x,y)=d(x,A)$. And if $X$ is a normed space that should be enough to prove $y in partial A$. Is there something beside this?
metric-spaces
asked 7 hours ago
Markus Steiner
735
One condition: if $Aneqvarnothing$ then $A$ should not be clopen. So this will not work in e.g. discrete metric space.
– drhab
6 hours ago
add a comment |
Let $(X,d)$ be a metric space, $A subseteq X$ and $x notin A$. What are the condition which make true the statement
$d(x,A) = d(x,partial A)$
If A is compact (not empty) I can think of a way to find a point $y in A$ such that $d(x,y)=d(x,A)$. And if $X$ is a normed space that should be enough to prove $y in partial A$. Is there something beside this?
metric-spaces
asked 7 hours ago
Markus Steiner
735
Let $(X,d)$ be a metric space, $A subseteq X$ and $x notin A$. What are the condition which make true the statement
$d(x,A) = d(x,partial A)$
If A is compact (not empty) I can think of a way to find a point $y in A$ such that $d(x,y)=d(x,A)$. And if $X$ is a normed space that should be enough to prove $y in partial A$. Is there something beside this?
metric-spaces
metric-spaces
asked 7 hours ago
Markus Steiner
735
asked 7 hours ago
Markus Steiner
735
asked 7 hours ago
Markus Steiner
735
asked 7 hours ago
Markus Steiner
735
asked 7 hours ago
Markus Steiner
735
735
One condition: if $Aneqvarnothing$ then $A$ should not be clopen. So this will not work in e.g. discrete metric space.
– drhab
6 hours ago
add a comment |
One condition: if $Aneqvarnothing$ then $A$ should not be clopen. So this will not work in e.g. discrete metric space.
– drhab
6 hours ago
One condition: if $Aneqvarnothing$ then $A$ should not be clopen. So this will not work in e.g. discrete metric space.
– drhab
6 hours ago
One condition: if $Aneqvarnothing$ then $A$ should not be clopen. So this will not work in e.g. discrete metric space.
– drhab
6 hours ago
add a comment |
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One condition: if $Aneqvarnothing$ then $A$ should not be clopen. So this will not work in e.g. discrete metric space.
– drhab
6 hours ago