ω* has (almost) no continuous images

Abstract

We prove that the following statement follows from the Open Colouring Axiom (OCA): if X is locally compact σ-compact but not compact and if itsČech-Stone remainder X * is a continuous image of ω * then X is the union of ω and a compact set. It follows that the remainders of familiar spaces like the real line or the sum of countably many Cantor sets need not be continuous images of ω * .

A separable non-remainder of ℍ

“…Comments on the construction. The proofs in [5,6] that certain spaces are not N * -images follow the same two-step pattern: first show that no 'trivial' map exists and then show that OCA implies that if there is a map at all, then there must also be a 'trivial' one. In the context of our example it should be clear that there is no map from H to the plane that induces a map from H * onto βK; it would have been nice to have found a map from l∈L I l+1 to the plane that would have induced h L , but we did not see how to construct one.…”

“…The final nail in the coffin of a putative map from H * onto the continuum K will be the following result from [5], where D = ω × (ω + 1).…”

Set theoretical aspects of the Banach space l∞/c0

“…and @A are not homeomorphic. A proof of this fact can be found in [3]. Below we shall shortly present another (obtained independently) argument of the claim.…”

Reduced Products of Metric Structures: A Metric Feferman–vaught Theorem

“…Commutative examples of such reduced products are well-known, for example under the Continuum Hypothesis C(βω \ ω) ∼ = C(βω 2 \ ω 2 ) (note that ℓ ∞ /c 0 ∼ = C(βω \ ω)), since by a well-known result of Parovičenko ([22]) under the Continuum Hypothesis βω\ω and βω 2 \ω 2 are homeomorphic. However under the proper forcing axiom they are not isomorphic (see [6] and [7,Chapter 4]). A naive way to obtain nontrivial isomorphisms, under the Continuum Hypothesis, between non-commutative coronas is by tensoring C(βω \ ω) and C(βω 2 \ ω 2 ) with a full matrix algebra.…”