Are Metric Space Completions Topologically Equivalent?

[Jamma]

Hi all,

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].

 

[Jamma]

Actually, think my intuition is wrong on this one, so ignore! (can't find "delete" button)

 

[SteveL27]

Actually, think my intuition is wrong on this one, so ignore! (can't find "delete" button)

Press Edit.

Then you get "Save/Go Advanced/Delete/Cancel" buttons. Press Delete.

You then get a "Do not Delete Message / Delete Message" pair of radio buttons. Select Delete. Then press "Delete this message." In other words you have to do four separate things to delete a message.

 

[Jamma]

I only have Save/Go Advanced/Cancel :/

Think that was true before, even with no other comments.

 

[blue_raver22]

I can provide some insights into the completion of a metric space. First, let's define what a metric space is - it is a set of points with a distance function that satisfies certain properties. The completion of a metric space is essentially extending this set of points to include all possible limits of Cauchy sequences, where a Cauchy sequence is a sequence of points whose terms become closer and closer together as the sequence progresses. This completion process is important because it allows us to fill in any "missing points" in the original metric space and make it a complete space, meaning that all Cauchy sequences in this new space will converge to a limit point within the space itself.

The two constructions mentioned in the content are different ways of achieving this completion. The first construction involves defining a pseudo-metric on all Cauchy sequences and then taking equivalence classes of these sequences to obtain the completion. The second construction involves taking the subspace of Cauchy sequences in the infinite product of the metric spaces and then identifying points with 0 distance as equivalence classes.

While these constructions may seem different, they actually lead to the same topological space. This is because in both cases, we are essentially identifying points that are "close" to each other in the original metric space and grouping them together as equivalence classes. In other words, the completion of a metric space is a way of "filling in the gaps" and making the space complete, and both constructions achieve this goal in a similar manner.

As for the concern raised in the content about the topological space of Cauchy sequences with pseudo-metric not being homeomorphic to the space of Cauchy sequences as a subspace of the infinite product, it is important to note that homeomorphism is a topological property that is preserved under continuous transformations. In this case, both constructions involve continuous transformations, so the resulting spaces will be homeomorphic.

In summary, the completion of a metric space is an important concept in mathematics and both constructions mentioned in the content lead to the same topological space. I can appreciate the elegance and efficiency of these constructions in filling in the gaps and making a metric space complete.