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