# How to Prove Isometries are Surjective on Compact Metric Spaces

• minibear
In summary, the conversation discusses how to show that two compact metric spaces with isometries have corresponding functions that are both surjective and inverse functions. It is suggested to use the theorem that states an isometry on a compact metric space must be surjective, and the result can be proven as a corollary of this.
minibear

## Homework Statement

Let (K, d) and (K', d') be two compact metric spaces and let f:K-->K' and g:K'--->K be isometries. Show that f(K)=K' and g(K')=K

n/a

## The Attempt at a Solution

I know that isometry implies that I can find one-to-one correspondence mapping, but not sure how to show both function and inverse function are subjective. Please help. Thanks!

minibear said:

## Homework Statement

Let (K, d) and (K', d') be two compact metric spaces and let f:K-->K' and g:K'--->K be isometries. Show that f(K)=K' and g(K')=K

## The Attempt at a Solution

I know that isometry implies that I can find one-to-one correspondence mapping, but not sure how to show both function and inverse function are subjective. Please help. Thanks!
I can't follow your attempt: we are not looking to 'find' a map. Could you please write out your work in full. It also helps to write out the definitions of the key terms: e.g., isometry, compact.

Are you aware of the theorem which says that an isometry $$h:X\to X$$ on a compact metric space X is surjective?

If so, the result you want to show is just a corollary of this. If not, then this is a good statement to prove.

## 1. What is the Isometry problem?

The Isometry problem is a mathematical problem that involves determining whether two geometric figures are congruent or not. In other words, it is the problem of determining whether two shapes are the same size and shape, regardless of their position or orientation in space.

## 2. What are some examples of isometric figures?

Some examples of isometric figures include cubes, spheres, and regular polyhedrons. These shapes have the same size and shape, but may be oriented differently in space.

## 3. How is the Isometry problem solved?

The Isometry problem can be solved using various methods, including coordinate geometry, transformations, and congruence postulates. These methods involve analyzing the properties and relationships of the given shapes to determine if they are congruent.

## 4. Why is the Isometry problem important?

The Isometry problem is important in various fields, including mathematics, engineering, and computer science. It helps in understanding and describing the properties of geometric figures, as well as in solving real-world problems involving spatial relationships.

## 5. Are there any real-life applications of the Isometry problem?

Yes, there are many real-life applications of the Isometry problem. For example, it is used in architecture and engineering to design structures that are symmetrical and stable. It is also used in computer graphics to create 3D models and animations, as well as in medical imaging to analyze and measure body structures.

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