Let $\Box\_{i\in I} X\_i$ denote the [box product](https://en.wikipedia.org/wiki/Box_topology) of the spaces $X\_i$. The box product $\Box\_{n\in\omega}\mathbb{R}$ is not connected, since the collection of bounded sequences is both open and closed. 


Is $\Box\_{n\in\omega}[0,1]$ connected?



Is $\Box_{n\in\omega}[0,1]$ connected?
It's not connected. 


Let $u=(u\_n)$ be a sequence. For $\ell\in [0,1]$, define $$V\_{u,\ell}=\{(v\_n):\forall n\in \omega:|v\_n-\ell|<\max(2^{-n},2|u\_n-\ell|)\}.$$
Then $u\in V\_{u,\ell}$ and $V\_{u,\ell}$ is open.


Moreover, if $\ell$ is a limit point of $u$ and $v\in V\_{u,\ell}$, then $\ell$ is a limit point of $v$.


Let $L\_\ell$ be the set of sequences in $[0, 1]$ converging to $\ell$, and $L\_\ell^c$ its complement. Then on the one hand, $L\_0=\bigcup\_{u\in L\_0}V\_{u,0}$, which is thus open. Its complement $L\_0^c$ is $\bigcup\_{u\in L\_0^c,\ell\in P(u,0)}V\_{u,\ell}$, where $P(u,\ell)$ is the set of limit points $\neq\ell$ of $u$. So it's open as well. Thus $L\_0$ is clopen. (Similarly $L\_\ell$ is open for every $\ell$.


