How can asymmetric intervals be generated by symmetric balls?

Question

Following this post: How can an interval be an open ball? There was an unresolved question in the comment

How can $(\alpha, \beta)$ be expressed in terms of open balls of the form $B(x, \epsilon) = (x - \epsilon, x+\epsilon)$, when the balls are symmetric and the intervals are not?

score 1 · Accepted Answer · answered Mar 22 '16 at 01:00

Write it like that :

$$]\alpha, \beta[ = \left] \frac{\beta + \alpha}{2} - \frac{\beta - \alpha}{2}, \frac{\beta + \alpha}{2} + \frac{\beta - \alpha}{2} \right[ = B(\frac{\beta + \alpha}{2} , \frac{\beta - \alpha}{2} )$$

ie. $x$ is the middle of $]\alpha, \beta[$

How can asymmetric intervals be generated by symmetric balls?

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