Let $U_x$ be a subset of $\mathcal P(S)$ such that $x \in \cap U_x$: with the following properties:
$(1)\,\,\,U_x$ is nested: for all $X,Y \in U_x$ either $X \subseteq Y$ or $Y \subseteq X$.
$(2)\,\,\,U_x$ is closed under unions
$(3)\,\,\,U_x$ is closed under non-empty intersections
Question: My question is whether the properties of $U_x$ alone are sufficient to define an equivalence relation---as in the example below---in order to induce a partition of $S$.
The kind of partition I have in mind can be exemplified on the assumption that $S$ is finite. If $S$ is finite then we can order its elements by natural numbers. So there exists an $n \in \mathbb N$ such that $U_x=\{X^0, X^1,...,X^n\}$ has the following form: $X^0 \subseteq X^1 \subseteq... \subseteq X^{n-1} \subseteq X^n$. This allows for the definition of the following equivalence relation on $S$: $\; y \sim_x z \;$ iff $\; \{y,z\}\subseteq X^k \setminus X^{k-1}$ for some $ 0\lt k\le n$. And the set $S/\sim_x$ is a partition of $S$.
Can this kind of partition be achieved in general, for any $S$ and $U_x$ with properties (1)-(3), or would some additional assumptions need to be made? By this kind I mean all elements of $U_x$ contributing in an orderly manner as in the given example.
Any help would be appreciated.
Consider $S = \Bbb R \cap [0, 1]$, $x = 1$. Define $X$ by $$ X = \{ [r, 1] \mid r \in S \}. $$
In other words, $X$ consists of all intervals starting at a point in $S$, and ending at $1$. This is closed under unions and under intersections.
But your proposed equivalence relation fails completely in this case, for there is no "previous" set to subtract from $X^r$, i.e., nothing analogous to $X^k \setminus X^{k-1}$.
In this case, Noah's partition consists of all singleton sets (which I proposed as an answer to your earlier question on this topic).