Please give an example of metric space that there are two open balls $B(x,\rho_1) \subset B(y,\rho_2)$ for $\rho_1>\rho_2$
How to construct this metric space
3
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real-analysis
functional-analysis
4 Answers
1
Take a discrete space with the discrete metric $d(x,x)=0$ and $d(x,y)=1$ for $x\ne y$. Then $B(x,1/3)=B(x,1/2)$.
2
If your $\subset$ allows the possibility of equality (i.e. $\subset$ means $\subseteq$), then letting $d$ be the discrete metric on any non-empty set $X$, we have that $B(x,3)=B(y,2)=X$ for any $x,y\in X$.
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0$\subset$ should allow eaqulity,I think :) – 2012-03-07
2
Let $X$ be the interval $(-3,3)$. Let $B_1 $ be the open ball of radius $3$ centered at $x=0$ and let $B_2$ be the open ball of radius 4 centered at $x=2$.
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0Thank you,it's so obvious by this! – 2012-03-07
1
The example is quite simple, take $X=[0,1]$ with the induced metric then take $B(0,10^6)\subset B(0,2)$.
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0I like to put a million in my calculus as well the number 13! – 2012-03-07