I am attempting to construct a partition of a countable set X, given a sigma-algebra on that set. Eventually I will want partitions of X to be in 1-1 correspondence with sigma-algebras on X. See my question below:

Say $X$ is countable. We want to show $\sigma$-algebras on $X$ are in one to one correspence with partitions of $X$.

Given a $\sigma$-algebra $\mathscr{A}$ on $X$, and given $x\in X$, we can form $$X_x=\bigcap_{A\in\mathscr{A}:x\in A}A\;.$$ This could be an uncountable intersection, meaning the result $X_x$ need not be an element of $\mathscr{A}$.

However, is there still a way to show these sets of the form $X_x$ do partition $X$? Or need I construct my partition in some other way?

Original image here.