Another isometric imbedding problem

[micromass]

Yes, now you've shown that every metric space has a completion. The usual way to prove this is by exercise 9. The reason most textbooks prefer exercise 9 is because it can be easily generalized and because the completion is very easy to describe.

One can in fact also show that the completion is unique. This is actually a consequence of exercise 2.

Also note that $$\mathbb{Q}$$ is an incomplete metric space. It's completion is of course $$\mathbb{R}$$. And exercise 9 now gives a very easy idea of how to construct $$\mathbb{R}$$! Just take all Cauchy sequences of rational numbers...

 

[radou]

Also note that $$\mathbb{Q}$$ is an incomplete metric space. It's completion is of course $$\mathbb{R}$$. And exercise 9 now gives a very easy idea of how to construct $$\mathbb{R}$$! Just take all Cauchy sequences of rational numbers...

Wow, I never thought of it that way! Thanks!

Btw, basically, this doesn't strictly have much to do with topology, right? i.e. it's more about metric spaces...

 

[micromass]

Yes, this is more analysis than topology. In fact, entire chapter 7 seems to be more about metric spaces than topology.

If you're going to study functional analysis, then you're going to see much of chapter 7 again. Specifically, the completion is very important in functional analysis! But you're correct, it's not really topology...