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

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.