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

Last revised on April 12, 2017 at 01:29:19. See the history of this page for a list of all contributions to it.