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A characterization of \(\omega_1\)-strongly countable-dimensional spaces in terms of \(K\)-approximations. (English) Zbl 1347.54051

A metrizable space \(X\) is 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 \(\omega_1\)-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 \(\text{dim}(X_\beta)<\infty\).
The 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 \(K\).

MSC:

54F45 Dimension theory in general topology
54E35 Metric spaces, metrizability
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