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 completeting 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).
Welcome! Math3ma is a math blog maintained by Tai-Danae, a PhD student at the CUNY Graduate Center.