## homeomorphism

Given sequences $(a_1, a_2, ...), (b_1, b_2, ...)$ of real numbers with $a_i > 0$ for all $i$, define $h: \mathbb{R}^\omega \rightarrow \mathbb{R}^\omega$ by the equation $h((x_1, x_2,...))=(a_1x_1+b_1, a_2x_2+b_2,...)$.
1. Show that if $\mathbb{R}^\omega$ is given the product toopology, $h$ is a homeomorphism.
2. What happens if $\mathbb{R}^\omega$ is given the box topology?
3. If $\mathbb{R}^\omega$ is given the uniform topology, under what conditions on the numbers $a_i, b_i$ is $h$ continuous? a homeomorphism?

For part 1, I know how to show $h$ is bijective, but having trouble showing $h$ and $h^{-1}$ continuous. I don't know how to do 2 and 3. Can I get some help please?