Metric and completeness of real numbers

• sunjin09
In summary, the conversation discusses the concepts of uncountability and completeness in metric spaces, and the relationship between these two notions. It is noted that completeness is a property of a metric space, while uncountability is a property of a set. The speaker also mentions using a distance d∈R that is not equal to any distances in a countable subspace to show that it is disconnected. The completion of R is mentioned as being necessary for defining a metric space, and the definition of open balls is discussed as being dependent on the uncountability of R. The speaker also clarifies that a metric can be defined before the completion of Q into R, but irrational radius open balls are only defined after completion and ordering of R.
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?

sunjin09 said:
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.

micromass said:
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?

sunjin09 said:
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.

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.

sunjin09 said:
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.

SteveL27 said:
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.

1. What is the metric of real numbers?

The metric of real numbers is the absolute value of the difference between two numbers. It measures the distance between two points on the real number line.

2. How is the completeness of real numbers defined?

The completeness of real numbers refers to the property that every non-empty set of real numbers that is bounded above has a least upper bound (also known as the supremum). In other words, there are no "gaps" in the real number line and every set of real numbers has a well-defined maximum value.

3. What is an example of a non-complete set of real numbers?

An example of a non-complete set of real numbers is the set of rational numbers. While the set is bounded above and below, there are certain numbers (such as the square root of 2) that are not included in the set and do not have a rational number approximation. Therefore, the set of rational numbers is not complete.

4. How does the completeness of real numbers relate to the decimal expansion of numbers?

The completeness of real numbers is closely related to the decimal expansion of numbers. The decimal expansion of a number is a representation of its value on the real number line. The completeness of real numbers ensures that every point on the real number line has a corresponding decimal expansion, without any gaps or missing values.

5. How is the completeness of real numbers useful in mathematics?

The completeness of real numbers is a fundamental concept in mathematics and has many important applications. It allows for the rigorous definition of limits, continuity, and convergence of sequences and series. It also allows for the use of techniques such as the intermediate value theorem and the Bolzano-Weierstrass theorem, which are essential in many areas of mathematics, such as calculus and analysis.

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