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

TL;DR

The paper constructs a compact space with specific reflection properties under set-theoretic assumptions, showing certain compactness properties do not reflect in small continuous images, and explores implications for Banach spaces.

Contribution

It provides a set-theoretic construction of a non-Corson compact space with all small continuous images Eberlein compacta, and analyzes reflection properties of functional tightness under Martin's axiom.

Findings

Existence of a compact space with non-reflecting properties under certain set-theoretic assumptions.

A Banach space of density ω₂ with all subspaces of density ω₁ being weakly compactly generated.

Countable functional tightness does not reflect in small continuous images under Martin's axiom.

Abstract

Assuming that there is a stationary set in $\omega_{2}$ of ordinals of countable cofinality that does not reflect, we prove that there exists a compact space which is not Corson compact and whose all continuous images of weight at most $\omega_1$ are Eberlein compacta. This yields an example of a Banach space of density $\omega_{2}$ which is not weakly compactly generated but all its subspaces of density $\omega_{1}$ are weakly compactly generated. We also prove that under Martin's axiom countable functional tightness does not reflect in small continuous images of compacta.