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open sets in this metric space

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Let E be an arbitrary set and, for p,q (elements of) E, define d(p,q) = 0 if p = q, d(p,q) = 1 if p does not equal q. This is a metric space. What are the open and closed balls in this metric space? Show that two balls of different centers and radii may be equal. What are the open sets in this metric space?

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The solution assess the open sets in this metric space.

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From the definition, for any p, q in E, either d(p,q) = 0 or d(p,q) = 1.
1. For each point p in E, the open ball B(p;r)={q in E: d(p,q)<r} depends on r. We have two cases.
Case 1: r<=1, then ...

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