Differences between the reduced Borel field and the category algebra of a space

Question

Let $X$ be a topological space.

Halmos calls "reduced Borel algebra" the quotient $B(X)/M(X)$ where $B(X)$ is the Borel field of $X$ and $M(X)$ is the $\sigma$-ideal of meagre subsets of $X$.

Fremlin calls "category algebra" the quotient $BP(X)/M(X)$ where $BP(X)$ is the Baire property algebra, i.e. the $\sigma$-field of subsets of $X$ having the Baire property (BP), and $M(X)$ is the sigma-ideal of meagre subsets of $X$.

I know that every Borel set has the BP (besides, it is an argument that Halmos uses to prove that $RO(X)$ (i.e. the regular open algebra of $X$) is isomorphic to $B(X)/M(X)$).

However, I wonder:

(1) What are the difference (if any) between $B(X)/M(X)$ and $BP(X)/M(X)$?

(2) If there are differences, how much does it depend on the topology of $X$ itself?

(3) Is it possible to prove that $RO(X)$ is isomorphic to $B(X)/M(X)$ without using the BP?

Many thanks in advance for any input and/or reference.

Answer 1

There is no difference, since every set with the Baire property is equivalent to an open set (thus Borel set) modulo the meager ideal.