On an example of neighborhood.

funcalys
Hi folks, as I was reviewing the metric space section in Amann- Escher textbook, I came across the following example of neighborhood:
"For $\left[0,1\right]$ with the metric induced from $R$, $\left[\frac{1}{2},1\right]$ is a neighborhood of 1, but not of $\frac{1}{2}$."
However I can't point out the exactly "r">0 satisfying $B_{[0,1]}(1,r)$$\subseteq[0,1]$.

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Won't any r < 1/2 do?

elgen
[1/2, 1] is a neighborhood of 1. In this case, r=1/2. Any element, of the ball with a radius of 1/2 centered at 1, has a distance less than 1/2 from 1.

[1/2,1] is not a neighborhood of 1/2. This is because any ball with a radius of r>0 centered at 1/2 contains some elements that are not in [1/2, 1].

funcalys
guess I misunderstood some of the concept in the first place, I thought the ball centered at 1 must completely lie in the interval [1/2,1].
:D.
Thank guys.