Spaces of continuous functions over a \(\Psi\)-space (Q2499784)

The authors of the present paper study the Lindelöf property of the space \(C_p(X)\) of continuous real-valued functions endowed with the topology of pointwise convergence. It is well known that (i) the Lindelöf property in the space \(C_p(X)\) is very rare and (ii) \(C_p(X)\) is Lindelöf if \(X\) is a separable metrizable space. Very recently R. D. Buzyakova discovered another class of spaces such that \(C_p(X)\) is Lindelöf. More precisely: ``The space \(C_p(X)\) is Lindelöf, if \(X=\alpha\cdot \{\beta\in\alpha: \text{cf}(\beta)> \omega\}\) for any ordinal \(\alpha\) with the usual ordinal topology''. The aim of the paper is to find other spaces, which are far from being metrizable and still have the property that the space \(C_p(X)\) is Lindelöf. For this purpose the authors are led to study \(\Psi\)- and \(\Psi\)-like spaces. Their main goal is to present two very interesting examples under the set-theoretical principle \(\diamondsuit\) with this property. These examples are also the main steps in answering some questions posed by Buzyakova.