Dyadicity index and metrizability of compact continuous images of function spaces

We show that the dyadicity index can be increased by taking the square even in the class of second countable spaces. Besides, any compact group contains a dense subspace of dyadicity index zero. We prove that, for any infinite cardinal κ, a compact space K with χ(x,K)⩾κ for any x∈K cannot be represented as a union of ⩽κ-many subspaces of network weight <κ. This fact has quite a few interesting consequences when we consider mappings of function spaces onto compact spaces. We prove, in particular, that if K is an ω1-monolithic Lindelöf Σ-space then every compact continuous image of Cp(K) is metrizable. For any cardinal κ an example is given of a compact space K such that Cp(K) maps continuously onto the Tychonoff cube of weight κ. We also establish that Luzin's axiom (2ω1>c) is equivalent to metrizability of all compact continuous images of Cp(K) whenever K is a separable compact space.