What are the components and path components of $\mathbb{R}^{\omega}$ in the product, uniform, and box topologies?

Question

I am working on an exercise problem about components and path components of $\mathbb{R}^{\omega}$. Specifically,

Exercise about components and path components:
1. What are the components and path components of $\mathbb{R}^{\omega}$ in the product topology?
2. What are the components and path components of $\mathbb{R}^{\omega}$ in the uniform topology?
3. What are the components and path components of $\mathbb{R}^{\omega}$ in the box topology?

I can only handle with only parts of the problem (about components):

My partial solution:
1. $\mathbb{R}^{\omega}$ in the product topology is connected, so its only component is $\mathbb{R}^{\omega}$.
2. $\mathbb{R}^{\omega}$ in the uniform topology is not connected (see here). There are two components: $A$ consisting of all bounded sequences of real numbers and $B$ of all unbounded sequences. [EDIT: I realized that the answer is wrong: $A$ and $B$ constitute a separation of $\mathbb{R}^{\omega}$ in the uniform topology. However, this does not imply that $A$ and $B$ are two components of it. So, I have no idea of this problem.]
3. No idea.

Therefore:

1. Is my partial solution correct?
2. How to figure out the other parts of the problem?