Hello math experts,
Let 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 .
I'm confused. Presumably you mean that . I'll help you with one half. Suppose that where is the set of all null sequences. Then, there exists some such that for every one has that there is some for which . So, let where if and otherwise. This is clearly open in the box topology since it's the product of open sets, but clearly . Thus is open etc. etc.