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A closure property for normal families of spaces. (English) Zbl 0787.54033

Summary: Let \(\mathcal P\) be a normal family of metrizable spaces with the property that for each member \(X\) of \(\mathcal P\) there exists a completely metrizable extension \(\widetilde{X}\) of \(X\) belonging to \(\mathcal P\). It is shown that \(\mathcal P\) has the following closure property: if \(K\) is a compact metric space such that for every \(\varepsilon > 0\) there exist an \(L\in {\mathcal P}\) and an \(\varepsilon\)-map \(f: K\to L\), then \(K\in {\mathcal P}\). This result has applications in the study of axiom systems for dimension theory.

MSC:

54F45 Dimension theory in general topology
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