On properties of compacta that do not reflect in small continuous images

Abstract

Abstract. Assuming that there is a stationary set of ordinals of countable cofinality in ω 2 that does not reflect, we prove that there exists a compact space which is not Corson compact and whose all continuous images of weight ≤ ω 1 are Eberlein compacta. This yields an example of a Banach space of density ω 2 which is not weakly compactly generated but all its subspaces of density ≤ ω 1 are weakly compactly generated.We also prove that under Martin's axiom countable functional tightness does not reflect in small continuous images of compacta.

Corson reflections

“…A discussion on why Lemma 3.4 does not apply to the space constructed in [18] is given in Example 4.3.…”

“…There exists a compact Hausdorff space X and an increasing sequence of elementary submodels M n ≺ H θ such that each M n splits X but n M n does not split X. This space is based on [18], and we include the relevant details for reader's convenience.…”

“…Since we were not aware of these results, the present paper should be considered as a survey rather than a research article.A compact Hausdorff space X is a Corson compactum (or shortly, Corson) if it is homeomorphic to a subspace of some Tychonoff cube [0, 1] κ which has the property that for every ξ < κ the set {x ∈ X : x(ξ) = 0} is countable.Every metrizable compactum is homeomorphic to a subspace of [0, 1] ω and therefore Corson. In [18] it was proved that if there exists a non-reflecting stationary subset of cofinality ω ordinals in ω 2 , then there exists a compact Hausdorff space X all of whose continuous images of weight ℵ 1 are Corson (and even uniform Eberlein; see §5), but X is not Corson.Theorem 1. Suppose κ is a supercompact cardinal.…”

“…Every metrizable compactum is homeomorphic to a subspace of [0, 1] ω and therefore Corson. In [18] it was proved that if there exists a non-reflecting stationary subset of cofinality ω ordinals in ω 2 , then there exists a compact Hausdorff space X all of whose continuous images of weight ℵ 1 are Corson (and even uniform Eberlein; see §5), but X is not Corson.…”

On Boolean algebras with strictly positive measures

“…Various aspects of (non)reflecting stationary sets were discussed in [11] and [1] and found several applications in topology and functional analysis, see e.g. [3], [15] and [12].…”