I'm trying to solve the following problem:
Considering the product topology on $\mathbb{R}^{\mathbb{R}}$. For each $A \subset \mathbb{R}$, let $\chi_A$ denote the characteristic function of $A$.
Let $\mathcal{F} = \{\chi_A : A \subset \mathbb{R}, A \text{ is countable } \} \subset \mathbb{R}^\mathbb{R}$. Show that $g \in \overline{\mathcal{F}}$, but there's no sequence on $\mathcal{F}$ such that $g$ is the limit of that sequence, where $g: \mathbb{R} \rightarrow \mathbb{R}$, $g(x) = 1$, i.e., the constant function 1.
I can't quite understand the $\mathcal{F} \subset \mathbb{R}^\mathbb{R}$ part; as far as I know, $\mathcal{F}$ is a family of functions, so how can a family of functions be considered a subset of a cartesian product? And how would I go about solving the problem? It would be great if someone could give me a hint of where to start.
