Revisit the problem " Image of open set through linear map "

Question

I am studying the following problem (2 years ago):

Image of open set through linear map

I have a few questions about the answer in that link:

1. What does the zero-neighborhood here mean? (I cannot find the definition in website)

2. What makes that proof different if $N$ is closed? ("$N$ is open" implies $N−x $ is a zero neighborhood?)

I guess the zero neighborhood means there is no neighborhood. If based on this, that answer makes no different if $N$ is closed to me.

Answer 1

1. A "zero neighborhood" is literally a neighborhood of zero, i.e. a set containing zero, that also contains an open set in which zero lies. In other words, we call $V$ a neighborhood of $x$ if $x\in V$ and there exists an open set $U\subset V$ such that $x\in U$. A zero neighborhood is the case where $x=0$ in a vector space.

2. The proof will not (immediately) work if you just assume $N$ to be any closed set. Note that a singleton set (with one element) is also a closed set, and so is a set with finitely many elements. So if you take $N=\{x\}$ then $N- \{x\}=\{0\}$, but this is not a neighborhood of anything, because it does not contain any non-trivial open set.