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 sequence $N_1, N_2, \ldots$ of nowhere dense subspaces (meaning that their closures have empty interiors) such that

$$A \cup \bigcup_i N_i = G \cup \bigcup_i N_i .$$

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.

category: foundational axiom