A subspace of a space is almost open if it is open modulo the -ideal of meagre subspaces?. We also say that has the Baire property.
Explicitly, is almost open if there exist an open subspace and an infinite sequences 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.
Created on June 8, 2010 04:06:22
by Toby Bartels