Let $X$ be a spectral space and $KO(X)$ be the set of all compact open subsets of $X$. Identify $KO(X)$ with $\{1_D:D\in KO(X)\}$, where $1_D(u) = 1$ if $u\in D$ and $1_D(u) = 0$ if $u\notin D$.
Question: Does the closure of $span {KO(X)}$ in $\ell^{\infty}(X)$ have a well-known equivalent description?
When $X$ is $T_1$, the closed linear span is $C(X)$, the space of all continuous functions on $X$.
