Is $\mathbf{R}^\omega$ in the uniform topology connected?
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.
The set of bounded sequences is both open and closed in this topology, so the space is disconnected.