# Continuity of a mapping in the uniform topology

In summary, the problem is to investigate the continuity of a map h : Rω --> Rω, defined as h((x1, x2, ...)) = (a1x1 + b1, a2x2 + b2, ...), under the condition that Rω is given the uniform topology. It was shown that h is a homeomorphism in the product topology, and since the uniform topology is finer than the product topology, it can also be concluded that h is continuous in the uniform topology. There does not seem to be any specific conditions on the numbers ai and bi, other than ai > 0, as stated in the problem. This reasoning can be generalized to state that, given a set X,
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## Homework Statement

Let (a1, a2, ...) and (b1, b2, ...) be sequences of real numbers, where ai > 0, for every i. Let the map h : Rω --> Rω be defined with h((x1, x2, ...)) = (a1x1 + b1, a2x2 + b2, ...). One needs to investigate under what conditions on the numbers ai and bi h is continuous, if Rω is given the uniform topology.

## The Attempt at a Solution

Now, in a previous exercise, it was shown that, if Rω (the set of all infinite sequences of real numbers) is given the product topology, h is a homeomorphism from of Rω with itself.

Further on, I know that the uniform topology is finer than the product topology.

Let x be a point in Rω, and h(x) its image. Let V be a neighborhood of h(x) in the product topology. V can be written as a union of basis elements from the product topology. Now, since the uniform topology is finer than the product topology, for every y in V we can find a basis element B of the uniform topology which is contained in the basis element of the product topology which contains y. Hence V equals the union of the basis elements B in the uniform topology.

Now, since h is continuous in the product topology, for V, there exists a neighborhood U of x such that h(U) is contained in V. Again, since the uniform topology is finer than the product topology, we can represent U as a union of basis elements from the uniform topology, and h(U) is still contained in V. Since this holds, we conclude that h is continuous in the uniform topology.

There doesn't seem to be any condition on the ai and bi' other than ai > 0, as stated in the problem formulation.

There's probably something wrong with my way of reasoning here, so please correct me if I'm wrong.

Actually, does this hold in general?

"Given a set X, a function f : X --> X, and two topologies on X, T and T', if f is continuous with respect to the topology T, and if T' is finer than T, then f is continuous with respect to T', too. "

Or, even more general?

"Given a continuous function f : X -- Y, where Tx and Ty are the topologies on X and Y, respectively, and given topologies Tx' and Ty' on X and Y, which are finer than Tx and Ty, respectively, f : X --> Y is continuous when X is given the topology Tx' and Y the topology Ty'."

## 1. What does it mean for a mapping to be continuous in the uniform topology?

In the uniform topology, a mapping is considered continuous if small changes in the input result in small changes in the output. This means that the pre-images of open sets in the output space are open in the input space.

## 2. How is continuity in the uniform topology different from continuity in other topologies?

In other topologies, continuity may be defined in terms of open sets or neighborhoods. However, in the uniform topology, continuity is defined in terms of the distance between points. This allows for a more general understanding of continuity.

## 3. Is continuity in the uniform topology the same as uniform continuity?

No, continuity in the uniform topology and uniform continuity are two different concepts. While continuity in the uniform topology relates to small changes in the input, uniform continuity relates to small changes in both the input and output.

## 4. How can continuity in the uniform topology be tested or proven?

One way to test continuity in the uniform topology is by using the epsilon-delta definition of continuity. This involves choosing an arbitrary small value (epsilon) and finding a corresponding small value (delta) such that the distance between the input and output points is less than epsilon whenever the distance between the input points is less than delta.

## 5. Why is continuity in the uniform topology important in mathematics?

Continuity in the uniform topology is important because it allows for a more general understanding of continuity. It also has applications in analysis, functional analysis, and topology, making it a fundamental concept in mathematics.

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