$R^ω$ is in the box topology. It is to be shown that $x$ and $y$ lie in the same component of $R^ω$ if and only if the sequence $x−y$ is “eventually zero.”
I tried to do it the following way but got stuck.
$(\Rightarrow)$ Suppose on the contrary that $x-y$ is not eventually zero. Then, define $U=\{z\in \mathbb R^\omega: x-z \text{ is eventually 0}\}$ and $V=\{z\in \mathbb R^\omega: x-z \text{ is not eventually 0}\}$. $U, V$ are non empty and disjoint; and $U\cup V=\mathbb R^\omega$. If $U$ and $V$ are shown to be open, then that would result in a contradiction, thus completing the proof.
So I tried to prove that $U$ is open. Take any $t\in U$. There exists $N$ such that for all $n>N$, $t_n=x_n$. But no matter what neighborhood of $t$, I take, it seems that none of the neighborhoods will contain only the sequences which coincide with $x$ eventually, thus giving a stronger reason to be believe that $U$ can't be open. Similarly, $V$ does not seem to be open either. So $U\cup V$ is probably not a separation of the given space. So this approach has probably failed. So how does one prove it now?
I have no idea about how to approach the converse.
Is there any 'not out of blue' approach to do this? Thanks.
Can anyone please recommend me a textbook on topology which has some solved problems of this type and also of the type 'topologist sine curve is connected not path connected'. Munkres' has the latter one but I find the proof there very complicated (i.e., I am able to follow the steps but I would forget it if it was asked to me say 2 days later) and it feels like memorizing the solution.
I also study from Topology without tears, I understand it but I didn't find exercises like the one in this post in the book. The book also talks about topologist sine curve being connected but not path connected but does so intuitively and I understand it but I couldn't write a proof of it on my own. I understand Munkres' book too but I get stuck at the exercises (now getting stuck and getting completely blank (like in the converse of the exercise in this post) are two different things and I tend to be on the latter side while doing the exercises in Munkres' and that's a problem). So can anyone please recommend me a book which has solved examples of various types so that I can try them out on my own and see the solution if I'm stuck? Thanks.
