Exercise · ZW-OPEN-BALL-009 · ★★★ Advanced
Let \(y\in B(a,r)\) and \(s=r-d(a,y)\). In \(\mathbb{R}\) with \(a=0\), \(r=5\) and \(y=3\), what is \(s\), the radius ensuring \(B(y,s)\subseteq B(a,r)\)?
Choose one option.
Show answer
- D. 2
Explanation
We get \(s=5-3=2\). By the triangle inequality, if \(d(y,z)<s\) then \(d(a,z)<d(a,y)+s=r\).
Review the concept: Open ball
The open ball \(B(x,r)\) with center \(x\) and radius \(r>0\) in a metric space \((X,d)\) is the set of points of \(X\) whose distance to \(x\) is less than \(r\).
\[B(x,r)=\{\,y\in X : d(x,y)
More practice problems: open ball
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- ZW-OPEN-BALL-006 In a space with the discrete metric, what is the ball B(a,1)?
- ZW-OPEN-BALL-007 In the usual ℝ, what is the intersection B(0,2)∩ B(3,2)?
- ZW-OPEN-BALL-008 How many points (a,b) with integer coordinates lie in the Euclidean ball B((0,0),3)?