# Questions on Completion of Metric Spaces and Isometries

• azdang
In summary, completing a discrete metric space X means ensuring that every Cauchy sequence in X converges to something in X itself. This is because the elements in a sequence must always be equal in order for the metric to be discrete. Since all cauchy sequences in a discrete space already converge, the completion of a discrete metric space is itself.
azdang
I actually have two questions I am having trouble with.

## Homework Statement

What is the completion of a discrete metric space X?

## Homework Equations

d(x,x) = 0
d(x,y) = 1 if x does not = y

I don't really understand how to complete a metric space that is incomplete. I just know that every Cauchy sequence in X would have to converge to something in X itself, but I'm not sure how to manipulate it to ensure that this happens.

AND

## Homework Statement

If X1 and X2 are isometric and X1 is complete, show that X2 is complete.

I know that since they are isometric, there is a mapping T such that d2(Tx,Ty) = d1(x,y). Other than that, I'm not sure how to prove it. It just kind of seems intuitive.

Any suggestions would be GREATLY appreciated. Thank you SO much.

What does a cauchy sequence look like in a discrete metric space? Look at the definition of cauchy and think about it.

Hmm...I'm not sure if this is right, but wouldn't you pick a sequence (xm) in the Discrete space that is Cauchy, then for every epsilon > 0 there exists N(epsilon) such that if m,n>N then d(xm, xn)< epsilon...but then, doesn't d(xm, xn)=1? But, how can we say that 1 < epsilon? I'm not sure, I really don't know what to do with this.

Pick epsilon=1/2. If d(xm,xn)<1/2, what does that tell you about xm and xn if the metric is discrete? The elements in a sequence don't HAVE to be different. But being in a discrete metric might force them to be the same. Hmm?

Ahh, so in order to complete this space, x and y must always be equal. That way, d(xm,xn) = 0 < epsilon. Am I getting that right?

You are getting closer. In order for a sequence to be cauchy, there must be an N such that xn=xm for all n,m>N. Agree? Does the sequence have a limit? What is it?

Okay, now I'm confused. Since x_m is NOT cauchy unless x_m = x_n, doesn't that say that every Cauchy sequence in the discrete space converges? So, isn't the discrete space already complete, and therefore, why would it need a completion?

azdang said:
Okay, now I'm confused. Since x_m is NOT cauchy unless x_m = x_n, doesn't that say that every Cauchy sequence in the discrete space converges? So, isn't the discrete space already complete, and therefore, why would it need a completion?

You don't sound confused to me. Yes, all cauchy sequences eventually repeat the same term over and over. Therefore all cauchy sequences converge to something in the space. Yes, discrete spaces are already complete. The completion is itself.

## 1. What is a metric space?

A metric space is a mathematical concept used to define the distance between two objects. It consists of a set of elements and a distance function that assigns a non-negative value to every pair of elements in the set. This distance function must also satisfy certain properties, such as the triangle inequality.

## 2. What does it mean for two metric spaces to be isometric?

Two metric spaces are isometric if there exists a bijective mapping between them that preserves the distance between any two elements. In other words, an isometry is a function that maintains the same distances between points in both metric spaces.

## 3. How is the completion of a metric space defined?

The completion of a metric space is the process of adding all the limit points of a given metric space to create a larger metric space in which the original space is a dense subset. This ensures that all Cauchy sequences in the original space converge to a limit in the completion.

## 4. What is the significance of isometries in metric spaces?

Isometries are important in metric spaces because they preserve important geometric properties, such as distances and angles. They also allow us to study the structure of a metric space by comparing it to a simpler isometric space.

## 5. How are isometries used in real-world applications?

Isometries have many practical applications, such as in computer graphics and computer-aided design (CAD). They are also used in physics to model the behavior of physical systems, and in economics and social sciences to analyze data and relationships between variables.

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