nLab
almost open subspace

A subspace $A$ of a space $X$ is almost open if it is open modulo the $\sigma$-ideal of meagre subspaces?. We also say that $A$ has the Baire property.

Explicitly, $A$ is almost open if there exist an open subspace $G$ and an infinite sequences $N_1,N_2,\dots$ of nowhere dense subspaces? (meaning that their closures have empty interiors) such that

$$A\cup \bigcup _i{N}_i=G\cup \bigcup _i{N}_i.$$A \cup \bigcup_i N_i = G \cup \bigcup_i N_i .

That every subspace of the real line is almost open follows from the axiom of determinacy? but contradicts the axiom of choice. In the absence of choice, it is a convenient assumption to make and is one of the axioms of dream mathematics.