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Given a metric space (X,d), one can take its completion by doing the following:

1) Take all Cauchy sequences of (X,d)

2) Define a pseudo-metric on these sequences by defining the distance between two sequences to be the limit of the termwise distance of the terms

3) Make this a metric by taking equivalence classes identifying any two sequences with 0 distance

4) This is your desired completion.

There is a slightly more topological approach which seems to mirror the above:

1) Take the (countable) infinite product of your metric spaces

2) Take the subspace (topology) of those points which are Cauchy sequences

3) Take the quotient space where we identify two points for which the Cauchy sequences are identified as above.

It feels as though these constructions should give rise to the same topological space, but I'm having troubles showing it - is there an immediately obvious reason that the two shouldn't be the same topologically?

[edit : to elaborate, my thoughts were that the topological space of Cauchy sequences with pseudo metric in the first construction won't be homeomorphic to the space of Cauchy sequences as a subspace of the infinite product, since things which "start off as distant in the first few terms" can converge to each other down the sequence. However, this might go away once we glue together points in the infinite product which have distance zero from each other as Cauchy sequences - very wordy, probably doesn't help, sorry].