$G_\delta$ ideals of compact sets

Abstract

Abstract. We investigate the structure of G δ ideals of compact sets. We define a class of G δ ideals of compact sets that, on the one hand, avoids certain phenomena present among general G δ ideals of compact sets and, on the other hand, includes all naturally occurring G δ ideals of compact sets. We prove structural theorems for ideals in this class, and we describe how this class is placed among all G δ ideals. In particular, we establish a result representing ideals in this class via the meager ideal. This result is analogous to Choquet's theorem representing alternating capacities of order ∞ via Borel probability measures. Methods coming from the structure theory of Banach spaces are used in constructing important examples of G δ ideals outside of our class.

Topologically invariant σ-ideals on the Hilbert cube

“…By Theorem 4.1 of [17], for every n ∈ ω there is a Tukey reduction g n : I n ∩ K(X) → NWD. The maps (g n ) n∈ω produce a Tukey reduction g = (g ω n ) n∈ω : n∈ω (I n ∩ K(X)) ω → NWD ω×ω .…”

Gδ Sets IN σ-IDEALS GENERATED BY COMPACT SETS

“…(3) for each n = 1, 2, 3, there exists a sequence of closed and upward closed sets (F n m ) m , such that the sets U = U n , V = V n and F m = F n m all satisfy (5), (6), (7), (8), and (9). To do this, fix a meager compact subset M of E, with at least three points, that is everywhere big with respect to I.…”

Tukey Order Among Ideals

“…We study the place F ideals of subsets of occupy in the Tukey order. As a point of reference we use the following, now standard, diagram which summarizes the known Tukey reductions among the well-studied directed sets; see [1,2,4,7,10,12,15,[17][18][19]20]. As usual, in the diagram, an arrow denotes reduction and the absence of arrows non-reduction.…”

“…Obviously, F : k J α k → Z 0 , and it is easy to check that it is a Tukey map. By (17), it suffices to show that, for appropriately chosen α k , k ∈ , we have…”