The Structure of σ-Ideals of Compact Sets

$G_\delta$ ideals of compact sets

“…The proposition below should be compared with a result due to Dougherty, Kechris, Louveau, and Woodin, [6], [8], that for I ⊆ K(E), I is an analytic σ -ideal if and only if I is a G δ ideal.…”

“…So (i) follows. If I is analytic, then it is a G δ by (i) and [8]. Assume now that it is coanalytic.…”

“…Of particular interest among definable ideals of compact sets are coanalytic σ -ideals mostly because of a wide range of examples belonging to this class and because of the theory that can be developed for it. By a dichotomy proved in [8], coanalytic σ -ideals fall into two major subclasses: they are either coanalytic complete or else they are G δ . This paper investigates the structure of the latter class of G δ σ -ideals of compact sets.…”

Calibrated thin $\boldsymbol {\Pi }_{\mathbf 1}^{\mathbf 1}$ $\sigma $-ideals are $\boldsymbol G_\delta $

“…Their results on the structure of σ-ideals of compact sets were used by Debs and Saint-Raymond ( [1]) to give a positive answer to the old question of whether every Borel U -set is meager. The main aim of this paper is to give an answer to an open question which was posed in [3], namely to prove the assertion from the title. Using our result we obtain a new proof of Kaufman's theorem concerning U -sets and U 0 -sets.…”

“…The theory of σ-ideals of compact sets was developed by Kechris, Louveau and Woodin in [3]. Their results on the structure of σ-ideals of compact sets were used by Debs and Saint-Raymond ( [1]) to give a positive answer to the old question of whether every Borel U -set is meager.…”