$[poly]: S \rightarrow \mathbb R;
[poly](x) \equiv
\begin{cases} 1 & x \in poly \\
0 & \text{otherwise} \end{cases}$

we can define a vector space of these functions over $\mathbb R$, using
the "scaling" action as the action of $\mathbb R$ on these functions:
The vector space $V$ is ```
*---* *-* *-* *
|###| |#| |#| |
|###| = |#| + |#| - |
|###| |#| |#| |
*---* *-* *-* *
```