4

Is $\mathbb R^\omega$ with box topology connected? Is it connected in uniform topology? It is connected with product topology. But I don't know the case for box and uniform topology.

Alex Ravsky
  • 106,166
Samiron Parui
  • 383
  • 1
  • 15
  • I think there is something wrong in your answer.it seems to me union of all such sets ${0}×⋯×{0}×\mathbb R×{0}…$ is not$\mathbb R^w$ but it is
    $\mathbb R^\infty$. Closure of${0}×⋯×{0}×$$ \mathbb R$$×{0}…$ is $R^w$ in product topology but not in box topology ..hence your argument is valid for product topology but not for box topology.     – Samiron Parui Nov 12 '17 at 06:11

1 Answers1

1

In order to mark the question as answered, I say that $\Bbb R^\omega$ is disconnected in the uniform topology, and so also in a stronger box topology. See this answer for more.

Alex Ravsky
  • 106,166