1. ## Box topology

Hello math experts,
Let $X=R^{N}$ be a topological space with the standard box topology.
Show that the collection of sequences that converge to 0 is an open-closed set in $X$.

thanks

2. Originally Posted by aharonidan
Hello math experts,
Let $X=R^n$ be a topological space with the standard box topology.
Show that the collection of sequences that converge to 0 is an open-closed set in $X$.

thanks
I'm confused. Presumably you mean that $X=\mathbb{R}^\mathbb{N}$. I'll help you with one half. Suppose that $(a_n)\notin Z$ where $Z$ is the set of all null sequences. Then, there exists some $\varepsilon>0$ such that for every $N\in\mathbb{N}$ one has that there is some $n\geqslant N$ for which $|a_n|>\varepsilon$. So, let $\displaystyle O=\prod_{n\in\mathbb{N}}A_n$ where $A_n=\mathbb{R}-(-\varepsilon,\varepsilon)$ if $|a_n|>\varepsilon$ and $\mathbb{R}$ otherwise. This is clearly open in the box topology since it's the product of open sets, but clearly $\displaystyle O\cap Z=\varnothing$. Thus $X-Z$ is open etc. etc.