The Lindelöf number of functional spaces on monolithic compacta (Q1791621)

Let \(X\) be a Hausdorff compact space and \(\tau\) an infinite cardinal. Denote by \(C_{c(\tau)}(X)\) the space of continuous real-valued functions on \(X\) with the topology of uniform convergence on compacta of weight \(\leq \tau\). The authors investigate the Lindelöf number \(\ell (C_{c(\tau)}(X))\) of this space. It is proved: (1) If \(X\) is \(\tau\)-monolithic and its tightness is \(\leq \tau\), then \(\ell(C_{c(\tau^+)}(X)) \leq \tau^+\); in particular, if the tightness of \(X\) is \(\leq \tau\), then \(\ell(C_{c(2^{\tau})}(X)) \leq 2^{\tau}\). (Recall that a space \(X\) is \(\tau\)-monolithic if for every \(A \subset X\) of cardinality \(\leq \tau\), the network weight of \(\overline{A}\) does not exceed \(\tau\)); (2) Under the assumption that no Aronszajn \(\tau^+\)-tree exists, \(\ell(C_{c(\tau)}(X)) \leq \tau\) whenever \(X\) is zero-dimensional, \(\tau\)-monolithic and its tightness is \(\leq \tau\).