Open balls in metric space property The 2019 Stack Overflow Developer Survey Results Are InProper inclusion between open ball, closure of open ball and the closed ball in a metric spaceOpen sets in a metric spaceThe union of open balls.Open Balls in Metric Space.Difference between open sets and open balls in metric spaceUnderstanding open balls in metric spacesabout shape of open ball in metric spaceIs the empty set an open ball in a metric space?Give a “constructive” proof of the fact that in a metric space the intersection of two open balls is openlimit in open set in metric space
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Open balls in metric space property
The 2019 Stack Overflow Developer Survey Results Are InProper inclusion between open ball, closure of open ball and the closed ball in a metric spaceOpen sets in a metric spaceThe union of open balls.Open Balls in Metric Space.Difference between open sets and open balls in metric spaceUnderstanding open balls in metric spacesabout shape of open ball in metric spaceIs the empty set an open ball in a metric space?Give a “constructive” proof of the fact that in a metric space the intersection of two open balls is openlimit in open set in metric space
$begingroup$
$textbfDefinition:$ Let $(X;d)$ a metric space $r>0$ and $a in X$. The open ball of center $a$ and radius $r$ is the set : $$ B(a;r)= x in X vert d(a,x)< r $$
And the closed ball of center $a$ and radius $r$ is the set : $$B[a;r]= xin X vert d(a,x)leq r $$
$textbfProblem$ Let $(X;d)$ a metric space, $ain X$ and $r>0$. Prove that :
$$ B(a;r)=bigcup_0<s<r B[a;s] $$
$textbfMi proof$
Let $x in B(a;r)$ then $d(x,a)<r$. Choose any $s>0$ such that : $0leq d(x,a)<s<r$.
$$ implies xin B(a;s) subseteq B[a;s] subseteq bigcup_0<s<r B[a;s]$$
Conversely $xin bigcup_0<s<r B[a;s]$ then exists $0<s<r$ such that $xin B[a;s]$
$$ implies d(a,x)leq s<r implies x in B(a;r) $$
I have two questions, one is if my proof is correct.
And the second is if it is true that:
$$ B(a;r)=bigcup_0<s<r B(a;s)$$
If it seems true, then in my test I have not used the inequality $leq$. Thanks!
metric-spaces
asked Apr 6 at 19:50
OrestedOrested
998
$endgroup$
add a comment |
$begingroup$
$textbfDefinition:$ Let $(X;d)$ a metric space $r>0$ and $a in X$. The open ball of center $a$ and radius $r$ is the set : $$ B(a;r)= x in X vert d(a,x)< r $$
And the closed ball of center $a$ and radius $r$ is the set : $$B[a;r]= xin X vert d(a,x)leq r $$
$textbfProblem$ Let $(X;d)$ a metric space, $ain X$ and $r>0$. Prove that :
$$ B(a;r)=bigcup_0<s<r B[a;s] $$
$textbfMi proof$
Let $x in B(a;r)$ then $d(x,a)<r$. Choose any $s>0$ such that : $0leq d(x,a)<s<r$.
$$ implies xin B(a;s) subseteq B[a;s] subseteq bigcup_0<s<r B[a;s]$$
Conversely $xin bigcup_0<s<r B[a;s]$ then exists $0<s<r$ such that $xin B[a;s]$
$$ implies d(a,x)leq s<r implies x in B(a;r) $$
I have two questions, one is if my proof is correct.
And the second is if it is true that:
$$ B(a;r)=bigcup_0<s<r B(a;s)$$
If it seems true, then in my test I have not used the inequality $leq$. Thanks!
metric-spaces
asked Apr 6 at 19:50
OrestedOrested
998
$endgroup$
4
$begingroup$
Your proof is correct and your claim is correct.
$endgroup$
– parsiad
Apr 6 at 19:57
$begingroup$
Thanks for your answer!
$endgroup$
– Orested
Apr 6 at 20:02
add a comment |
$begingroup$
$textbfDefinition:$ Let $(X;d)$ a metric space $r>0$ and $a in X$. The open ball of center $a$ and radius $r$ is the set : $$ B(a;r)= x in X vert d(a,x)< r $$
And the closed ball of center $a$ and radius $r$ is the set : $$B[a;r]= xin X vert d(a,x)leq r $$
$textbfProblem$ Let $(X;d)$ a metric space, $ain X$ and $r>0$. Prove that :
$$ B(a;r)=bigcup_0<s<r B[a;s] $$
$textbfMi proof$
Let $x in B(a;r)$ then $d(x,a)<r$. Choose any $s>0$ such that : $0leq d(x,a)<s<r$.
$$ implies xin B(a;s) subseteq B[a;s] subseteq bigcup_0<s<r B[a;s]$$
Conversely $xin bigcup_0<s<r B[a;s]$ then exists $0<s<r$ such that $xin B[a;s]$
$$ implies d(a,x)leq s<r implies x in B(a;r) $$
I have two questions, one is if my proof is correct.
And the second is if it is true that:
$$ B(a;r)=bigcup_0<s<r B(a;s)$$
If it seems true, then in my test I have not used the inequality $leq$. Thanks!
metric-spaces
asked Apr 6 at 19:50
OrestedOrested
998
$endgroup$
$textbfDefinition:$ Let $(X;d)$ a metric space $r>0$ and $a in X$. The open ball of center $a$ and radius $r$ is the set : $$ B(a;r)= x in X vert d(a,x)< r $$
And the closed ball of center $a$ and radius $r$ is the set : $$B[a;r]= xin X vert d(a,x)leq r $$
$textbfProblem$ Let $(X;d)$ a metric space, $ain X$ and $r>0$. Prove that :
$$ B(a;r)=bigcup_0<s<r B[a;s] $$
$textbfMi proof$
Let $x in B(a;r)$ then $d(x,a)<r$. Choose any $s>0$ such that : $0leq d(x,a)<s<r$.
$$ implies xin B(a;s) subseteq B[a;s] subseteq bigcup_0<s<r B[a;s]$$
Conversely $xin bigcup_0<s<r B[a;s]$ then exists $0<s<r$ such that $xin B[a;s]$
$$ implies d(a,x)leq s<r implies x in B(a;r) $$
I have two questions, one is if my proof is correct.
And the second is if it is true that:
$$ B(a;r)=bigcup_0<s<r B(a;s)$$
If it seems true, then in my test I have not used the inequality $leq$. Thanks!
metric-spaces
metric-spaces
asked Apr 6 at 19:50
OrestedOrested
998
asked Apr 6 at 19:50
OrestedOrested
998
asked Apr 6 at 19:50
OrestedOrested
998
asked Apr 6 at 19:50
OrestedOrested
998
asked Apr 6 at 19:50
OrestedOrested
998
998
4
$begingroup$
Your proof is correct and your claim is correct.
$endgroup$
– parsiad
Apr 6 at 19:57
$begingroup$
Thanks for your answer!
$endgroup$
– Orested
Apr 6 at 20:02
add a comment |
4
$begingroup$
Your proof is correct and your claim is correct.
$endgroup$
– parsiad
Apr 6 at 19:57
$begingroup$
Thanks for your answer!
$endgroup$
– Orested
Apr 6 at 20:02
4
4
$begingroup$
Your proof is correct and your claim is correct.
$endgroup$
– parsiad
Apr 6 at 19:57
$begingroup$
Your proof is correct and your claim is correct.
$endgroup$
– parsiad
Apr 6 at 19:57
$begingroup$
Thanks for your answer!
$endgroup$
– Orested
Apr 6 at 20:02
$begingroup$
Thanks for your answer!
$endgroup$
– Orested
Apr 6 at 20:02
add a comment |
0
active
oldest
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4
$begingroup$
Your proof is correct and your claim is correct.
$endgroup$
– parsiad
Apr 6 at 19:57
$begingroup$
Thanks for your answer!
$endgroup$
– Orested
Apr 6 at 20:02