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 $N_1,N_2,\ldots$ 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.

Created on June 8, 2010 04:06:22
by Toby Bartels
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