Completing a Metric Space, Intuitively

An incomplete metric space is very much like a golf course: it has a lot of missing points! (A golf course has exactly eighteen "missing points" a.k.a. holes.) The process of completing a metric space is akin to filling the holes ( = limits of Cauchy sequences) in a golf course ( = the metric space) with dirt ( = equivalence classes of Cauchy sequences).

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