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

Assuming that there is a stationary set of ordinals of countable cofinality in ω₂ that does not reflect, we prove that there exists a compact space which is not Corson compact and whose all continuous images of weight ≤ω₁ are Eberlein compacta. This yields an example of a Banach space of density ω₂ which is not weakly compactly generated but all its subspaces of density ≤ω₁ are weakly compactly generated. We also prove that under Martin's axiom countable functional tightness does not reflect in small continuous images of compacta.