2

Let $\mathbb{R}^w$ denote the space of sequences in $\mathbb{R}$ and consider it with the uniform topology, i.e. with the metric

$d(x,y) = \sup\limits_{i\ge 1} \{ \min(|x_i - y_i|), 1)\}$

Determine if $\mathbb{R}^w$ is connected.

My claim that it is not. Can someone verify my proof?

Attempt

I claim that $A$ and $B$ for a separation of $\mathbb{R}^w$, where $A$ denotes the set of unbounded sequences and $B$ the set of bounded sequences. It is clear that $A \cup B = \mathbb{R}^w$ and that $A \cap B = \emptyset$. So it only remains to show that $A$ and $B$ are open. But that is clear as well, since any $\epsilon-$ball with $\epsilon < 1$ centered at a bounded/unbounded sequence must remain bounded/unbounded too. Hence $A$ and $B$ are open and $\mathbb{R}^w$ is not connected.

JustANoob
  • 1,729
  • 11
  • 25
  • 2
    Looks good to me. Why were you worried about the validity? – Dan Rust Aug 05 '22 at 12:47
  • 2
    See https://math.stackexchange.com/q/300173/977780 – SoG Aug 05 '22 at 12:50
  • @DanRust Because it is very easy to convince one self that ones proof is correct. I was searching for this eact question before I posted and could not find it. It appeared after i posted it though :) – JustANoob Aug 06 '22 at 16:04

0 Answers0