This is from Apostol's Calculus, Vol. II Section 8.3 #3(a).
Prove that $S=\{(x,y,z) \mid z^2 - x^2 - y^2 -1>0\}$ is open.
The only way to prove that a set is open which has been covered so far is to prove that for an arbitrary point $\mathop a\limits^{\small \to} \in S$ there is an open ball $B(\mathop a\limits^{\small \to} , r) \subset S$ by finding an explicit $r$ and showing that $\mathop x\limits^{\small \to} \in B(\mathop a\limits^{\small \to} , r) \implies \mathop x\limits^{\small \to} \in S$. Nothing topological (compactness, completeness, etc.) has been discussed. For some reason I just am stuck trying to figure out an $r$ which would work.
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