almost open subspace

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

Explicitly, AA is almost open if there exist an open subspace GG and an infinite sequence N 1,N 2,N_1, N_2, \ldots of nowhere dense subspaces (meaning that their closures have empty interiors) such that

A iN i=G iN 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.

Revised on April 12, 2017 01:29:19 by Toby Bartels (