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...

@KennyLau you're right ! the sum was meant to be outside the fraction. about Y though it doesn't matter what Y is I just want a function so that the inverse image of Z will give me V. and I think that Z is that set $(0,\infty)\cup(0,\infty)$ — excalibirr, Jul 31 '18 at 04:18
take $Y$ to be two points with the discrete topology. map $V$ to one point and $M \setminus V$ to the other point. — Kenny Lau, Jul 31 '18 at 04:19
so what would that look like put into notation.for example we could say a continuous function f:\Bbb R^2 \rightarrow \Bbb R, (x,y)\rightarrow xy. how would we right the function you have suggested in this way ? — excalibirr, Jul 31 '18 at 04:22
@KennyLau the indiscrete topology has only two members...I'm not sure if you understand what im asking here. I want to find a continous function f so that i can use the fact that Z is open or closed to prove that f^{-1}(Z)=V is open/closed — excalibirr, Jul 31 '18 at 04:27
I don't quite understand what the question says. Are $Y$ and $Z$ at our disposal? Why can't you take $Y=M$, $f$ the identity map and $Z=V$? — Kavi Rama Murthy, Jul 31 '18 at 06:15
@KaviRamaMurthy Yes Y and Z are at our disposal. I don't know exactly why you couldn't do exactly what you're saying but it doesn't sound right (it seems too easy..) I'm no expert so I can't say why though.I just basically want to use the theorem which states that if f:X\rightarrow Y is a continuous function and Z\cupY is open/closed Then f^{-1}(Z) is open/closed to show that V is open or closed as was done here https://math.stackexchange.com/questions/2867600/finding-a-continuous-function-to-show-that-this-set-is-open-or-closed#2867608 — excalibirr, Jul 31 '18 at 07:24
@exodius I don't understand where the question of $V$ being closed comes in. It is not required that $Z$ is closed so its inverse image also need not be closed. I don't think the question has anything to do with any set being closed. I think my example is correct. — Kavi Rama Murthy, Jul 31 '18 at 07:28
@KaviRamaMurthy The only thing I'm trying to show is that V is open or closed... — excalibirr, Jul 31 '18 at 07:29
@exodius I now see that point in your title but you should have included that in the body of the question. Please state what you want carefully in future. — Kavi Rama Murthy, Jul 31 '18 at 07:31

score 1 · Accepted Answer · answered Jul 31 '18 at 05:14

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.

answered Jul 31 '18 at 05:14

copper.hat

172,524

would you mind elaborating a little bit ? 1. are you saying this is not an answer because you haven't used a continuous function to show this 2. Is it impossible to find a continuous function because it is both open and closed ?3. would your answer gain me full marks on an exam ? – excalibirr Jul 31 '18 at 05:19
1

Well, 1. I'm not sure if $V$ is the set you had in mind, if it is then it is neither open not closed. 2. If $V$ is the set in question, then it is not clear how you could use the inverse function of an open/closed set to show what you want. 3. If the question is to show that $V$ is neither open nor closed then my answer is a valid proof, however your examiner may have other considerations so I can't really answer 3. – copper.hat Jul 31 '18 at 05:23

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

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