Density properties and points of uncountable order for families of open sets in function spaces
Let \(X\) be a Tychonoff space and \(C_p(X)\) denote the space of all continuous real valued functions on \(X\) with the topology of pointwise convergence. The aim of the present work is to study the relation between various cardinal functions on \(X\) and those on \(C_p(X)\). For each subset \(A\subset X\) and each cardinal \(\kappa\) let \([A]_\kappa= \cup \{\overline B:B \subset A\) and \(|B|\leq \kappa\}\). We define the tightness \(t(X)= \min\{\kappa: \overline A=[A]_\kappa\) for all \(A\subset X\}\). We say that \(A\) is \(\kappa\)-dense in \(X\) in case \([A]_\kappa=X\). We define the density tightness \(dt(X)=\min \{\kappa\): every dense subset of \(X\) is \(\kappa\)-dense in \(X\}\). The authors show that \(dt(C_p(X))= t(C_p(X))\) for any \(X\). For an infinite cardinal \(\mu\) we say that \(X\) is \((\kappa,\mu)\)-compact in case, for every subset \(\{x_\alpha: \alpha\in A\} \subset X\) with \(|A|=\kappa\), there is an accumulation point \(x\in X\) such that, if \(U\) is a neighborhood of \(x\), then \(|\{\alpha: x_\alpha\in U\} |\geq\mu\). Among their other results is a characterization of the \((\kappa,\mu)\)-compactness of finite powers of \(X\) in terms of the topology of \(C_p(X)\). A number of unsolved problems are also posed.
scientific article; zbMATH DE number 1787573