Metric and completeness of real numbers

[sunjin09]

So far I have learned a bit about topological spaces, there has been several occasions regarding metric spaces where I had to invoke completeness (or at least uncountability) of real numbers R, which is itself a property of the usual metric space of R. For example, to show that any countable subspace of a metric space is disconnected, I had to come up with a distance d$\in$R such that d isn't equal to any distances in the countable space. So I assume the completion of R must come before a metric may even be defined, which is a mapping into R; the definition of an open ball is even more dependent, since it is a bijective mapping from (X, R) to the set of all open balls in X. Does this sound about right?

 

[micromass]

So far I have learned a bit about topological spaces, there has been several occasions regarding metric spaces where I had to invoke completeness (or at least uncountability) of real numbers R, which is itself a property of the usual metric space of R.

Uncountability and completeness are two different notions. Completeness is a notion which is dependent on a metric. Uncountability is just a notion dependent on a set. You don't need a metric space to take about countable.

For example, to show that any countable subspace of a metric space is disconnected, I had to come up with a distance d$\in$R such that d isn't equal to any distances in the countable space.

You don't need completeness of R to solve this. You may want to use that R is uncountable however.

So I assume the completion of R must come before a metric may even be defined, which is a mapping into R;

Huh?

the definition of an open ball is even more dependent, since it is a bijective mapping from (X, R) to the set of all open balls in X.

An open ball is a bijective mapping? I'm sorry, I'm not following you.

 

[sunjin09]

Uncountability and completeness are two different notions. Completeness is a notion which is dependent on a metric. Uncountability is just a notion dependent on a set. You don't need a metric space to take about countable.



You don't need completeness of R to solve this. You may want to use that R is uncountable however.

I agree.

Huh?

d(x,y) maps into R, which has already been completed with irrationals, although d need not be irrational.

An open ball is a bijective mapping? I'm sorry, I'm not following you.

An open ball is defined as B_ε(x0)={x | d(x,x0)<ε}. So ψ: (X,R)→{set of all open balls in X} is surjective mapping where ψ(x0,ε)=B_ε(x0), not bijective, sorry. I guess it is only the uncountability that matters, but the uncountability come from completion of Q into R, doesn't it?

 

[micromass]

An open ball is defined as B_ε(x0)={x | d(x,x0)<ε}. So ψ: (X,R)→{set of all open balls in X} is surjective mapping where ψ(x0,ε)=B_ε(x0), not bijective, sorry.

Oh, that's what you mean! Yes, that's right then. But note that there are spaces with only a finite or countable number of open balls!

I guess it is only the uncountability that matters, but the uncountability come from completion of Q into R, doesn't it?

Yes, R is uncountable as a consequence of completeness. So in some sense, you need completeness for the problem.

 

[sunjin09]

So I guess the metric can be defined before Q is completed into R, but then the open balls' radii are all rational by definition. After completion and ordering of R, irrational radius open balls can then be defined. That sounds like logical. Thank you for clarification.

 

[SteveL27]

So I guess the metric can be defined before Q is completed into R, but then the open balls' radii are all rational by definition. After completion and ordering of R, irrational radius open balls can then be defined. That sounds like logical. Thank you for clarification.

You know, that's an interesting point. A metric by definition is a function from pairs of elements of a metric space, into the reals. So when we start with the rationals, we can define the distance between rationals, but we hold off on defining open balls until we've constructed the reals as the completion of the rationals.

I've never thought about this before but I believe you're right. You need to define the reals before defining a metric space.

But in practice it's not a problem. You can define the distance between rationals as the absolute value of their difference, and you can define Cauchy sequences. So there's no logical problem.

 

[sunjin09]

But in practice it's not a problem. You can define the distance between rationals as the absolute value of their difference, and you can define Cauchy sequences. So there's no logical problem.

Yes, the completion of R need not refer to the notion of a metric space. A rational distance is sufficient to define Cauchy sequences.