The behaviour of dimension functions on unions of closed subsets
Let \(d\) be a dimension function. For ordinals \(\beta, \alpha\) such that \(\beta<\alpha\), and a space \(X\) of a class \(\mathcal K\) of topological spaces with \(dX=\alpha\), the authors define the values \(m(X,d,\beta,\alpha)=\min\{k:X=\bigcup_{i=1}^kX_i\), where \(X_i\) is closed in \(X\) and \(dX_i\leq\beta\}\), \(m_{\mathcal K}(d,\beta,\alpha)=\min\{m(X,d,\beta,\alpha): X\in{\mathcal K}\) and \(m(X,d,\beta,\alpha)\) exists\(\}\) and \(M_{\mathcal K}(d,\beta,\alpha)=\sup\{m(X,d,\beta,\alpha): X\in{\mathcal K}\) and \(m(X,d,\beta,\alpha)\) exists\(\}\). In this paper, they determine the values of \(m_{\mathcal P}(\)Cmp,\(\beta,\alpha)\), \(M_{\mathcal P}(\)Cmp,\(\beta,\alpha)\), \(m_{\mathcal C}(\)trInd,\(\beta,\alpha)\) and \(M_{\mathcal C}(\)trInd,\(\beta,\alpha)\), where \({\mathcal P}\) and \({\mathcal C}\) mean the classes of separable completely metrizable spaces and metrizable compact spaces, respectively. Furthermore, they introduce new dimension functions: the additive compactness degree Cmp\(_{\cup}\) and the transfinite additive inductive dimension functions ind\(_{\cup}\) and Ind\(_{\cup}\), and discuss the values \(m_{\mathcal P}(\)Cmp\(_{\cup},\beta,\alpha)\), \(M_{\mathcal P}(\)Cmp\(_{\cup},\beta,\alpha)\), \(m_{\mathcal C}(\)ind\(_{\cup},\beta,\alpha)\), \(m_{\mathcal C}(\)Ind\(_{\cup},\beta,\alpha)\), \(M_{\mathcal C}(\)ind\(_{\cup},\beta,\alpha)\) and \(M_{\mathcal C}(\)Ind\(_{\cup},\beta,\alpha)\).
transfinite additive inductive dimension
scientific article; zbMATH DE number 2110893