A characterization of \(\omega_1\)-strongly countable-dimensional spaces in terms of \(K\)-approximations (Q2788948)

A metrizable space \(X\) is \textit{locally finite-dimensional } if for every \(x\in X\), there exists an open set \(U\subset X\) such that \(x\in U\) and \(\dim(U)<\infty\). The space \(X\) is \textit{\(\omega_1\)-strongly countable-dimensional} if there exists an ordinal \(\alpha<\omega_1\) such that \(X=\bigcup\{X_\beta: \beta<\alpha\}\) where every \(X_\beta\) is an open subset of the space \(X\setminus(\bigcup \{X_\gamma: \gamma<\beta\})\) and \(\text{dim}(X_\beta)<\infty\).NEWLINENEWLINEThe author gives a characterization for a space \(X\) to be \(\omega_1\)-strongly countable-dimensional (or locally finite-dimensional) in terms of existence of \(K\)-approximations for mappings of \(X\) into an arbitrary metric simplicial complex \(K\).