Question

Let S be the set of all real circles, defined by (x-a)^2 + (y-b)^2=r^2. Define d(C1,C2)...

Let S be the set of all real circles, defined by (x-a)^2 + (y-b)^2=r^2. Define d(C1,C2) = √((a1 − a2)^2 + (b1 − b2)^2 + (r1 − r2)^2 so that S is a metric space.

Prove that metric space S is NOT a complete metric space. Give a clear example.

Describe points of C\S as limits of the appropriate sequences of circles, where C is the completion of S.

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