Dyadicity index and metrizability of compact continuous images of function spaces
All spaces are supposed to be Tychonoff. The authors note that in [Russ. Math. Surv. 35, 145--156 (1980; Zbl 0462.54013)], \textit{B. E. Shapirovsky} defined \textit{the dyadicity index} id\((X)\) for an arbitrary space \(X\) to be the minimal cardinal \(\kappa\) such that \(X\) cannot be continuously mapped onto \([0,1]^{\kappa^+}\). And they note that in the preceding article a number of deep results were presented that concern dyadicity index and its relationships with \(\pi\)-character and tightness. The authors show that the dyadicity index can be increased by taking the square even in the class of second countable spaces. They prove that, for any infinite cardinal \(\kappa\), a compact space \(K\) with \(\chi(x,K)\geq\kappa\) for any \(x\in K\) cannot be represented as a union of \(\leq\kappa\)-many subspaces of network weight \(<\kappa\). The authors prove that if \(K\) is an \(\omega_1\)-monolithic Lindelöf \(\Sigma\)-space then every compact continuous image of \(C_p(K)\) is metrizable. For any cardinal \(\kappa\) an example is given of a compact space \(K\) such that \(C_p(K)\) maps continuously onto the Tychonoff cube of weight \(\kappa\). They also prove that Luzin's axiom (\(2^{\omega_1}>\mathfrak c\)) is equivalent to the metrizability of all compact continuous images of \(C_p(K)\) whenever \(K\) is a separable compact space. They conclude by presenting a number of open problems concerning dyadicity index properties and the metrizability of compact continuous images of ``nice'' function spaces.
scientific article; zbMATH DE number 2173008
scientific article