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

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