TITLE: Is $\mathbf{R}^\omega$ in the uniform topology connected?
QUESTION [16 upvotes]: Let $\mathbf{R}^\omega$ be the set of all (infinite) sequences of real numbers. Then is this space connected in the uniform topology? How to determine this? 
The uniform metric $p \colon \mathbf{R}^\omega \times  \mathbf{R}^\omega \to \mathbf{R}$ is defined as follows: $$p((x_n),(y_n)) := \sup_{n\in Z^+} \min\{|x_n-y_n|,1\}$$ for sequences $(x_n)$, $(y_n)$ of real numbers.

REPLY [26 votes]: The set of bounded sequences is both open and closed in this topology, so the space is disconnected.