Corson reflections

Abstract

A reflection principle for Corson compacta holds in the forcing extension obtained by Levy-collapsing a supercompact cardinal to ℵ2. In this model, a compact Hausdorff space is Corson if and only if all of its continuous images of weight ℵ1 are Corson compact. We use the Gelfand-Naimark duality, and our results are stated in terms of unital abelian C * -algebras.Before starting, we should thank Alan Dow for pointing our attention to [2]. Our use of C * -algebras is closely related to Bandlow's use of large Hilbert cubes. Similar methods have been used in[15], [11], and it is possible that the C * -algebraic vantage point may yield additional applications. A paper of Kunen ([15]) contains a closely related analysis of Corson compact spaces. Although our main theorem is not a logical consequence of Bandlow's and Kunen's results, most of the ideas are contained in their papers. 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. Then the following reflection statement holds in V Coll(ℵ 1 ,<κ) : If X is a compact Hausdorff space, then all continuous images of X of weight at most ℵ 1 are Corson compact if and only if X is Corson. The same principle follows from Martin's Maximum.