(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Here's another problem from Munkres.

Let (a1, a2, ...) and (b1, b2, ...) be sequences of real numbers, with ai > 0, for every i. Define h : Rω --> Rω with h((x1, x2, ...)) = (a1x1 + b1, a2x2 + b2, ...). Show that if Rω is given the product topology, h is a homeomorphism.

2. Relevant equations

I used a theorem which states that if f : A --> ∏Xj is given by the equation f(a) = (fj(a)) (j is from some indexing set J), where fj : A --> Xj, for each j, then f is continuous if and only if fj is continuous, for each j.

I'm not sure if I can use this theorem here, since there's no information about what the set A is.

3. The attempt at a solution

First I tried to prove that h is continuous using the theorem up there.

h(x) can be represented with (h1(x), h2(x), ...), where hj(x) = aj∏j(x) + bj. Here ∏j(x) denotes the projection mapping onto the j-th coordinate of x. Since hj(x) is continuous for every j (the projection mapping is continuous, addition and multiplication are, too), we conclude that h is continuous.

Proving one to one and onto is easy:

Let h(x) = h(y), then for every j, ajxj + bj = ajyj + bj, hence xj = yj, for every j. Let c = (c1, c2, ...) be in the image set of h. Then, for every j, (cj - bj)/aj = xj, and hence (x1, x2, ...) maps to c.

The j-th component of the inverse of h is given with (hj(x) - bj)/aj, and the inverse is continuous too, since it's continuous for every index j.

All this is well defined, since aj > 0, for every j.

Hence, h is a homeomorphism from R∞ to R∞.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Homeomorphism problem

**Physics Forums | Science Articles, Homework Help, Discussion**