  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.
