Topology on $\mathbb{R}$ Induced by Lebesgue Density: Regular but Not Normal?

Question

Let a neighborhood basis of a point $x$ of the real line consist of all Lebesgue-measurable sets containing $x$ whose density at $x$ equals $1$. Show that this requirement defines a topology that is regular but not normal.

Some thoughts

I believe with this definition of topology is simply the discrete topology of $\mathbb{R}$. In fact, unless our definition of local base differs, $[x,x+2]$ is an open neighbourhood of $x+1$ and $[x-2,x]$ an open neighborhood of $x-1$. Hence, $\{x\}$ is open as intersection of two open sets.

Maybe, we are referring to the topology whose base consists in all Lebesgue measurable sets $E$ such that any point of $E$ is a density point of $E$?

This problem was shared with me by my friend. Neither of us has a solution.Any solutions would be appreciated.