# Conservation of completeness by uniformly continuous bijection

• h-simplex
In summary, the conversation discusses the proposition that a uniformly continuous bijection between metric spaces, where one is complete, implies that the other is also complete. The attempt at a solution includes a partial proof that hinges on the claim that the inverse of a uniformly continuous bijection is also uniformly continuous. The speaker suggests trying to find counterexamples for both statements.
h-simplex

## Homework Statement

I want to prove this proposition:

Let $f: M \rightarrow N$ be a uniformly continuous bijection between metric spaces. If M is complete, then N is complete.

## The Attempt at a Solution

I have a 'partial' solution, whose legitimacy hinges upon a claim that I am unable to prove, namely that the inverse of a uniformly continuous bijection is uniformly continuous.

Assuming that is true, here goes my 'proof':
Let $(y_n)$ be a Cauchy sequence in N. Then, since $f^{-1}$ is uniformly continuous, $(f^{-1}(y_n))$ is Cauchy in M; since M is complete, there exists $a \in M$ such that $f^{-1}(y_n) \rightarrow a$. Hence by continuity of f, we have $f(f^{-1}(y_n)) = y_n \rightarrow f(a) \in N$. Thus N is complete.

The above proof seems very 'natural' ... but as I mentioned, it hinges on something I am unable to prove, or disprove. So is my claim correct and if so how can I prove it? If not... well then I have no idea how to prove the original claim; this was my best idea.

I would try to find a counterexample for both statements (that is: "uniform continuous images of complete spaces are complete" and "uniform continuous bijections are uniform isomorphisms"). I doubt that they are true.

## 1. What is the concept of conservation of completeness by uniformly continuous bijection?

The concept of conservation of completeness by uniformly continuous bijection is a mathematical theorem that states that a uniformly continuous bijection between two complete metric spaces preserves completeness. This means that if a metric space is complete, any uniformly continuous bijection from that space to another space will also be complete.

## 2. How does a uniformly continuous bijection preserve completeness?

A uniformly continuous bijection preserves completeness by ensuring that all Cauchy sequences in the original metric space have a corresponding Cauchy sequence in the target metric space. This is possible because the uniformly continuous function ensures that the distance between points in the original space is maintained in the target space.

## 3. Why is the concept of conservation of completeness important in mathematics?

The concept of conservation of completeness is important in mathematics because it allows for the extension of properties and theorems from one metric space to another through a uniformly continuous bijection. This allows for the study of more complex spaces by utilizing the properties of simpler, complete spaces.

## 4. Can a non-uniformly continuous bijection also preserve completeness?

No, a non-uniformly continuous bijection cannot preserve completeness. This is because a non-uniformly continuous function can stretch or compress distances between points, which can result in Cauchy sequences in the original space not having corresponding Cauchy sequences in the target space.

## 5. Are there any real-world applications of conservation of completeness by uniformly continuous bijection?

Yes, there are real-world applications of conservation of completeness by uniformly continuous bijection in fields such as physics and engineering. For example, the concept is used in the study of fluid dynamics and the design of structures to ensure that distances and properties are preserved between two different spaces.

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