Finding a net that converges to the zero map in $\mathbb{R}^\mathbb{R}$

Question

I'm looking through Willard's General Topology and I came across this problem in the section on nets.

Willard 11A.1 In $\mathbb{R}^{\mathbb{R}}$, let $E = \{f \in \mathbb{R}^{\mathbb{R}} \mid f(x) = 0$ or $1$ and $f(x) = 0$ only finite often$\}$ and let $g$ be the function in $\mathbb{R}^{\mathbb{R}}$ which is identically 0. Then, in the product topology on $\mathbb{R}^{\mathbb{R}}$, $g \in \overline{E}$. Find a net $(f_\lambda)$ in $E$ which converges to $g$.

If I'm understanding this correctly, we need to find a net of $\mathbb{R}$-tuples that converge to the zero map $g$. And it seems each member of the net needs to come from $E$, which means the net must consist of these $1$-$0$ valued strings. And to keep things from becoming trivial, we are only allowed for a finite number of $0$'s to occur in each member of the net. Otherwise, you could just choose the zero map.

Can I receive some support or assistance on how to approach this problem? I'm not sure where to go on this.

Answer 1

For every $A\subseteq\mathbb R$ we can use the indicator function (or characteristic function) as $$\chi_A(x)= \begin{cases} 1,&\text{if }x\in A;\\ 0,&\text{if }x\notin A. \end{cases} $$ We will work with the functions $f_A(x)=1-\chi_A(x)$. (But I used this in the definition, since the characteristic function is something you might encounter quite often.)

Let $D$ be the set of all finite subsets of $\mathbb R$.

Try to verify that:

• $(D,\subseteq)$ is a directed set.
• For every $A\in D$, the function $f_A$ belongs to the set $E$.
• The net $(f_A)_{A\in D}$ converges to the constant function $0$.

Answer 2

If you agree that $g$ is in the closure of $E$, then automatically $g$ is a limit of a net with values in $E$. That's a general result in any topological space. Namely, you can take the set of neighborhoods of $g$ (ordered by reverse inclusion) as the index set for the net (you need to show it's a directed set) and for each such nbhd pick an element of $E$ contained in it. That defines a net which converges to $g$.