NECESSARY AND SUFFICIENT CONDITION for $x$ and $y$ lying in the same component of $R^{\omega}$ in uniform topology.

Question

Consider $R^{\omega}$ in the uniform topology. Show that $x$ and $y$ lie in the same component of $R^{\omega}$ if and only if the sequence $x - y = ({x_1 -y_1 , x_2 - y_2, .......})$ is bounded.

How to prove this?

My Try : By uniform topology We can not have a separation in $R^{\omega}$. Because we can not have separation on $R$.

I don't know I am wrong or right. Any help will be highly appreciated. Thank You.

Answer 1

No no I think I was wrong. However I got the answer here. I hope It would also help the reader. $R^{\omega}$ is disconnected in Uniform and Box topology.

You must check this out -What are the components and path components of $\mathbb{R}^{\omega}$ in the product, uniform, and box topologies?