Help constructing a continuous function to prove whether a set is open or closed

Question

I have the space $M=\Bbb R^{\Bbb N}$

the distance function $d(a,b)=\sum\frac{2^{-n}|a_n-b_n|}{1+|a_n-b_n|}$

and the subset $V=\{a \in M|\{n \in \Bbb N|a_n \neq 0\}finite\}$.

I'm trying to find a continuous function $f:M\rightarrow Y$ s.t. $Z\subset Y$ and $f^{-1}(Z)=V$

I think the set would be $(-\infty,0) \cup(0,\infty)$ but i can't figure out how to construct the function itself...

Answer 1

Not an answer.

If $a \in V$ then let $a_n(k)= \begin{cases} a(k),& k \le n \\ 1, & \text{otherwise}\end{cases}$. Note that $a_n \notin V$ and $a_n \to a$. Hence $V$ is not open.

Let $e(k) = 1$ for all $k$, and let $e_n(k)= \begin{cases} 1,& k \le n \\ 0, & \text{otherwise}\end{cases}$. Then $e_n \in V$ for all $n$ and $e_n \to e$, but $e \notin V$. Hence $V$ is not closed.