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

Assuming that there is a stationary set in $ω_{2}$ 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 $ω_1$ are Eberlein compacta. This yields an example of a Banach space of density $ω_{2}$ which is not weakly compactly generated but all its subspaces of density $ω_{1}$ are weakly compactly generated. We also prove that under Martin's axiom countable functional tightness does not reflect in small continuous images of compacta.

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