# Injective map from sigma algebras to partitions

I want to show that for every sigma-algebra $\mathfrak A$, you can define for all $x\in X$ the set of all $A_x:=\bigcap_{A\in \mathfrak A,x\in A}A$( a partition of the set $X$) and that this map is injective. Thus, for different sigma-algebras we get different partitions. Does anybody have a good hint how to start with this?

This is not true. Look at the Borel sets on $[0,1]$ and the power set on $[0,1]$ At each point, this intersection is the point itself. They are clearly different $\sigma$-algebras.