Let $X$ be a compact metric space, and consider the Banach space $\Omega = C(X,\mathbb R)$ of continuous, real-valued functions on $X$, equipped with the supremum norm. Let $\delta_x \in \Omega^*$ be the evaluation functional, defined by $\delta_x(f) = f(x)$. For any compact $D \subseteq X$, consider the $\sigma$-algebra $$\mathcal F_D = \bigcap_{\epsilon > 0} \sigma \{ \delta_x : x \in D^\epsilon\},$$ where $D^\epsilon$ denotes the $\epsilon$-enlargement of the set $D$ in $X$. Loosely speaking, the $\sigma$-algebra $\mathcal F_D$ encodes the information stored in (an infinitesimal neighborhood of) the set $D$.

The family $\mathbb F_0 = \{\mathcal F_D\}$ admits the obvious partial ordering ($\mathcal F_D \preceq \mathcal F_{D'}$ if $D \subseteq D'$), and moreover is a lattice. It is also a partially-ordered, right-continuous filtration.

**Question 1:** What is the general structure of the family $\mathbb F_0$?

**Question 2:** What is the general structure of the space $\mathbb F$ of *all* $\sigma$-algebras over $\Omega$? Is this space too big to have any nice structure? Is there a more appropriate space to call "the space of $\sigma$-algebras"?