# How to construct a partition of X given a sigma-algebra on X (when X countably infinite) [duplicate]

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.