# Homework Help: Continuity of a mapping in the uniform topology

1. Sep 27, 2010

1. The problem statement, all variables and given/known data

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.

3. 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.

2. Sep 27, 2010

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'."